Deck 2: Propositional Logic: Syntax and Semantic

Question 1

use the following key to determine which of the translations of the given English argument to PL is best.
B: Brouwer is an intuitionist.
F: Frege is a logicist.
G: Gödel is a platonist.
H: Hilbert is a formalist.
-It is not the case that either Frege is a logicist or Brouwer is an intuitionist. Gödel being a platonist is necessary and sufficient for Brouwer being an intuitionist. Hilbert is a formalist. So, Gödel is not a platonist; however, Hilbert is a formalist.

A) ∼F $\lor$ B
G ≡ B
H / ∼G • H
B) ∼F $\lor$ B
G ≡ B
H / ∼G ≡ H
C) ∼(F $\lor$ B)
G ≡ B
H / ∼G • H
D) ∼(F $\lor$ B) G ≡ B
H / ∼G ⊃ H
E) ∼(F $\lor$ B) G ≡ B
H / H ⊃ ∼G

Answer 1

C

Question 2

use the following key to determine which of the translations of the given English argument to PL is best.
B: Brouwer is an intuitionist.
F: Frege is a logicist.
G: Gödel is a platonist.
H: Hilbert is a formalist.
-Hilbert is a formalist if, and only if, Gödel is a platonist. Hilbert is not a formalist and Brouwer is an intuitionist. Hilbert is a formalist if Frege is a logicist. Therefore, Frege is not a logicist and Gödel is not a platonist.

A) H ≡ G ∼H • B
H ⊃ F / ∼F • ∼G
B) H ≡ G ∼H • B
F ⊃ H / ∼F • ∼G
C) G ⊃ H ∼H • B
F ⊃ H / ∼F • ∼G
D) H ⊃ G ∼H • B
F ⊃ H / ∼F • ∼G
E) H ⊃ G ∼H • B
H ⊃ F / ∼F • ∼G

Answer 2

B

Question 3

use the following key to determine which of the translations of the given English argument to PL is best.
B: Brouwer is an intuitionist.
F: Frege is a logicist.
G: Gödel is a platonist.
H: Hilbert is a formalist.
-If Gödel is a platonist, then Frege is a logicist. If Frege is a logicist, Brouwer being an intuitionist is a sufficient condition for Hilbert being a formalist. Gödel is a platonist. Gödel is a platonist if Hilbert is a formalist. Therefore, Gödel is a platonist if Brouwer is an intuitionist.

A) G ⊃ F
F ⊃ (B ≡ H)
G
H ⊃ G / B ⊃ G
B) G ⊃ F
F ⊃ (B ≡ H)
G
G ⊃ H / G ⊃ B
C) G ⊃ F
F ⊃ (H ⊃ B)
G
H ⊃ G / B ⊃ G
D) G ⊃ F F ⊃ (B ⊃ H)
G
H ⊃ G / B ⊃ G
E) G ⊃ F F ⊃ (B ⊃ H)
G
G ⊃ H / G ⊃ B

Answer 3

D

Question 4

use the following key to determine which of the translations of the given English argument to PL is best.
B: Brouwer is an intuitionist.
F: Frege is a logicist.
G: Gödel is a platonist.
H: Hilbert is a formalist.
-If Frege is a logicist, then Brouwer is an intuitionist. If Brouwer is an intuitionist, then Gödel is a platonist only if Hilbert is a formalist. Gödel is a platonist. Frege is a logicist. So, Hilbert is a formalist.

A) F ⊃ B B ⊃ (H ⊃ G)
G
F / H
B) F ⊃ B B ⊃ (G ⊃ H)
G
F / H
C) F ⊃ B B ⊃ (G ≡ H)
G
F / H
D) F ⊃ B (B ⊃ G) ⊃ H
G
F / H
E) F ⊃ B (B ⊃ H) ⊃ G
G
F / H

Question 5

use the following key to determine which of the translations of the given English argument to PL is best.
B: Brouwer is an intuitionist.
F: Frege is a logicist.
G: Gödel is a platonist.
H: Hilbert is a formalist.
-If Frege is a logicist and Brouwer is an intuitionist, then Hilbert is a formalist and Gödel is a platonist. Hilbert is not a formalist. Brouwer is an intuitionist. Either Frege is a logicist or both Gödel is not a platonist and Brouwer is an intuitionist. Therefore, Gödel is not a platonist.

A) F • [B ⊃ (H • G)]
∼H
B
F $\lor$ (∼G • B) /∼G
B) (F • B) ⊃ (H ⊃ G)
∼H
B
(F $\lor$ ∼G) • B /∼G
C) (F • B) ⊃ (H • G)
∼H
B
F $\lor$ (G • B) / ∼G
D) (F • B) ⊃ (H • G)
∼H
B
(F $\lor$ ∼G) • B / ∼G
E) (F • B) ⊃ (H • G)
∼H
B
F $\lor$ (∼G • B) / ∼G

Question 6

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-D $\lor$ ∼E

A) If Dewey denigrated the quest for certainty, then Peirce did not emphasize education.
B) If Dewey denigrated the quest for certainty, then Peirce emphasized education.
C) Either Dewey denigrated the quest for certainty or Peirce emphasized education.
D) Either Dewey denigrated the quest for certainty or Peirce did not emphasize education.
E) Dewey denigrated the quest for certainty unless Peirce emphasized education.

Question 7

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-∼(A • E)

A) It is not the case that Peirce either studied logic or emphasized education.
B) It is not the case that Peirce both studied logic and emphasized education.
C) Peirce neither studied logic nor emphasized education.
D) Peirce did not both not study logic and not emphasize education.
E) Peirce did not study logic and James was not a pluralist.

Question 8

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-∼C ⊃ (A ≡ B)

A) If Dewey wrote about thirdness, then Peirce studied logic just in case James was a pluralist.
B) If Dewey did not write about thirdness, then Peirce studying logic is a necessary condition for James to be a pluralist.
C) Dewey did not write about thirdness provided that Peirce studied logic if, and only if, James was a pluralist.
D) Dewey not writing about thirdness entails Pierce studying logic and James being a pluralist.
E) If Dewey did not write about thirdness, then Peirce studied logic if, and only if, James was a pluralist.

Question 9

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-[∼D • (∼E ⊃ ∼C)]

A) Both Dewey did not denigrate the quest for certainty and Peirce did not emphasize education, given that Dewey did not write about thirdness.
B) Dewey not denigrating the quest for certainty and Peirce not emphasizing education are necessary conditions for Dewey not writing about thirdness.
C) Dewey did not denigrate the quest for certainty just in case if Peirce did not emphasize education then Dewey did not write about thirdness.
D) Dewey did not denigrate the quest for certainty if, and only if, Peirce not emphasizing education entails Dewey not writing about thirdness.
E) Dewey did not denigrate the quest for certainty and Dewey did not write about thirdness given that Peirce did not emphasize education.

Question 10

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-(A ≡ B) ⊃ C

A) Peirce studying logic is necessary and sufficient for James being a pluralist, given that Dewey wrote about thirdness.
B) If Peirce studied logic just in case James was a pluralist, then Dewey did not write about thirdness.
C) If Peirce studied logic if, and only if, James was a pluralist, then Dewey wrote about thirdness.
D) If Peirce studying logic is necessary for James being a pluralist, then Dewey wrote about thirdness.
E) If James being a pluralist is necessary for Peirce studying logic, then Dewey wrote about thirdness.

Question 11

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-∼(C $\lor$ D)

A) It is not the case that Dewey writing about thirdness entails his denigrating the quest for certainty.
B) Dewey did not both write about thirdness and denigrate the quest for certainty.
C) Dewey neither wrote about thirdness nor denigrated the quest for certainty.
D) Either Dewey did not write about thirdness or Dewey did not denigrate the quest for certainty.
E) Dewey not writing about thirdness entails his not denigrating the quest for certainty.

Question 12

use the following key to determine which English sentence best represents the given formula of PL.
A: Peirce studied logic.
B: James was a pluralist.
C: Dewey wrote about thirdness.
D: Dewey denigrated the quest for certainty.
E: Peirce emphasized education.
-∼B ⊃ (∼D • ∼E)

A) If James was not a pluralist, then neither Dewey denigrated the quest for certainty nor Peirce emphasized education.
B) Dewey did not denigrate the quest for certainty and Peirce did not emphasize education given that James was not a pluralist.
C) James not being a pluralist entails neither Dewey denigrated the quest for certainty nor Peirce did not emphasize education.
D) James was not a pluralist provided that Dewey did not denigrate the quest for certainty and Peirce did not emphasize education.
E) If James was not a pluralist, then either Dewey did not denigrate the quest for certainty or Peirce did not emphasize education.

Question 13

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-∼A $\lor$ (B • D)

A) It's a wff. The main operator is the ∼.
B) It's a wff. The main operator is the $\lor$ .
C) It's a wff. The main operator is the •.
D) Not a wff

Question 14

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-I • E ⊃ ∼F

A) It's a wff. The main operator is the •.
B) It's a wff. The main operator is the ⊃.
C) It's a wff. The main operator is the ∼.
D) Not a wff

Question 15

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-{∼R ⊃ [(Q • P) ⊃ S]}

A) It's a wff. The main operator is the ∼.
B) It's a wff. The main operator is the first ⊃, reading left to right.
C) It's a wff. The main operator is the •.
D) It's a wff. The main operator is the second ⊃, reading left to right.
E) Not a wff

Question 16

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-[(P ≡ Q) • ∼Q] ⊃ (P ⊃ R)

A) It's a wff. The main operator is the ≡.
B) It's a wff. The main operator is the •.
C) It's a wff. The main operator is the ∼.
D) It's a wff. The main operator is the first ⊃, reading left to right.
E) Not a wff

Question 17

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-[A $\lor$ B • C] ⊃ (A $\lor$ C)

A) It's a wff. The main operator is the first $\lor$ , reading left to right.
B) It's a wff. The main operator is the •.
C) It's a wff. The main operator is the ⊃.
D) It's a wff. The main operator is the second $\lor$ , reading left to right.
E) Not a wff

Question 18

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-P ⊃ (Q ⊃ R) ⊃ [(P • ∼R) ⊃ ∼Q]

A) It's a wff. The main operator is the first $\lor$ , reading left to right.
B) It's a wff. The main operator is the ≡.
C) It's a wff. The main operator is the second $\lor$ , reading left to right.
D) It's a wff. The main operator is the •.
E) Not a wff

Question 19

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-[(D ⊃ ∼E) • (F ⊃ E)] ⊃ [D ⊃ (∼F $\lor$ G)]

A) It's a wff. The main operator is the first ⊃, reading left to right.
B) It's a wff. The main operator is the second ⊃, reading left to right.
C) It's a wff. The main operator is the third ⊃, reading left to right.
D) It's a wff. The main operator is the fourth ⊃, reading left to right.
E) Not a wff

Question 20

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-∼{[(H ⊃ I) ⊃ ∼(I $\lor$ ∼J)] ⊃ (∼H ⊃ J)}

A) It's a wff. The main operator is the first ∼, reading left to right.
B) It's a wff. The main operator is the ≡.
C) It's a wff. The main operator is the second $\lor$ , reading left to right.
D) It's a wff. The main operator is the •.
E) Not a wff

Question 21

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-∼[(R • S) ⊃ U] ⊃ {{∼U ⊃ [R ⊃ (S ⊃ T)]}

A) It's a wff. The main operator is the first ⊃, reading left to right.
B) It's a wff. The main operator is the first ∼, reading left to right.
C) It's a wff. The main operator is the second ⊃, reading left to right.
D) It's a wff. The main operator is the third ⊃, reading left to right.
E) Not a wff

Question 22

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator.
-[(W ⊃ X) • (Y $\lor$ ∼X)] ≡ [∼(Z $\lor$ Y) ⊃ ∼W]

A) It's a wff. The main operator is the •.
B) It's a wff. The main operator is the ≡.
C) It's a wff. The main operator is the first ∼, reading left to right.
D) It's a wff. The main operator is the second ⊃, reading left to right.
E) Not a wff

Question 23

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼B ⊃ Y

A) True
B) False
C) Unknown

Question 24

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼X ≡ A

A) True
B) False
C) Unknown

Question 25

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-X $\lor$ [A • (B ⊃ Y)]

A) True
B) False
C) Unknown

Question 26

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-X $\lor$ [A • (Y ⊃ B)]

A) True
B) False
C) Unknown

Question 27

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼Y ⊃ [A ≡ (Y • B)]

A) True
B) False
C) Unknown

Question 28

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-X ⊃ [(∼X $\lor$ A) ⊃ X]

A) True
B) False
C) Unknown

Question 29

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(Z $\lor$ ∼A) ≡ [(A $\lor$ ∼Z) ⊃ (X ≡ ∼X)]

A) True
B) False
C) Unknown

Question 30

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-[(Y ⊃ ∼Y) ⊃ (B ⊃ ∼B)] ⊃ [( B $\lor$ Y) ≡ (∼B $\lor$ ∼Y)]

A) True
B) False
C) Unknown

Question 31

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼{∼[(∼A $\lor$ ∼X) • ∼A] • ∼X}

A) True
B) False
C) Unknown

Question 32

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-{X $\lor$ [C • (Y ⊃ B)]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z)]}

A) True
B) False
C) Unknown

Question 33

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-Q • (∼A ≡ Q)

A) True
B) False
C) Unknown

Question 34

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-Q • (∼A • ∼Q)

A) True
B) False
C) Unknown

Question 35

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼Q $\lor$ (∼X $\lor$ Q)

A) True
B) False
C) Unknown

Question 36

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(C $\lor$ X) ⊃ (Q $\lor$ A)

A) True
B) False
C) Unknown

Question 37

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-[(P ⊃ X) ⊃ P] ⊃ P

A) True
B) False
C) Unknown

Question 38

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(Y $\lor$ P) ⊃ (B • P)

A) True
B) False
C) Unknown

Question 39

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(∼P ⊃ P) $\lor$ (A ⊃ P)

A) True
B) False
C) Unknown

Question 40

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(∼P ⊃ P) $\lor$ (P ⊃ A)

A) True
B) False
C) Unknown

Question 41

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼(Q ⊃ C) $\lor$ (Z • ∼X)

A) True
B) False
C) Unknown

Question 42

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼[(Z ⊃ B) • (P ⊃ C)] $\lor$ [(X • Y) ≡ A]

A) True
B) False
C) Unknown

Question 43

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(P ⊃ ∼Q) $\lor$ ∼P

A) True
B) False
C) Unknown

Question 44

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-(P ≡ Q) $\lor$ (P ≡ ∼Q)

A) True
B) False
C) Unknown

Question 45

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-[(P $\lor$ Q) ⊃ X] ≡ ∼(P $\lor$ Q)

A) True
B) False
C) Unknown

Question 46

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-[(P • Q) $\lor$ (∼P • Q)] $\lor$ [(P • ∼Q) $\lor$ (∼P • ∼Q)]

A) True
B) False
C) Unknown

Question 47

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression.
-∼[(A ⊃ P) $\lor$ (A ⊃ Q)] • (P • Q)

A) True
B) False
C) Unknown

Question 48

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(B ? B) $\lor$ ?B

A) 11
B) 10
C) 01
D) 00

Question 49

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(D ? ?D) ? (?D ? D)

A) 11
B) 10
C) 01
D) 00

Question 50

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-?[F • (F ? F)]

A) 11
B) 10
C) 01
D) 00

Question 51

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-U ? (Z ? U)

A) 1100
B) 0011
C) 0001
D) 1110

Question 52

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(I ? J) $\lor$ (J ? I)

A) 1111
B) 1100
C) 1010
D) 0000

Question 53

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(K ? L) ? (?K ? L)

A) 1111
B) 1010
C) 0110
D) 1000
E) 0001

Question 54

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(M $\lor$ N) • (M $\lor$ ?N)

A) 1111
B) 1110
C) 1010
D) 1100
E) 1000

Question 55

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(V ? W) • (V ? ?W)

A) 1111
B) 1001
C) 0110
D) 1000
E) 0000

Question 56

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(?X • Y) ? (X • Y)

A) 1111
B) 1101
C) 1011
D) 1000
E) 0010

Question 57

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-A ? [(?B $\lor$ A) ? ?A]

A) 1100
B) 1001
C) 0011
D) 0001
E) 0000

Question 58

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(?C ? ?D) ? [D ? (C ? D)]

A) 1111
B) 1101
C) 1011
D) 0011
E) 0101

Question 59

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[E $\lor$ ?(E • ?F)] ? [F ? ?(E $\lor$ ?F)]

A) 1111
B) 1000
C) 0010
D) 0111
E) 0000

Question 60

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-G ? {(H ? ?G) ? [(G • ?H) $\lor$ (H • ?G)]}

A) 1111
B) 1100
C) 0111
D) 0100
E) 0000

Question 61

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[J ? (K $\lor$ L)] ? [?L ? (K $\lor$ J)]

A) 1111 1111
B) 1111 1110
C) 1110 1111
D) 1111 1100
E) 0101 0101

Question 62

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[M $\lor$ (N • O)] $\lor$ [(M • N) $\lor$ (?M • N)]

A) 1111 1111
B) 1100 1100
C) 1111 1100
D) 1100 0000
E) 0000 1100

Question 63

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[P ? (?R ? Q)] $\lor$ [(Q $\lor$ R) $\lor$ (?P • R)]

A) 1111 1111
B) 1110 1111
C) 1110 1110
D) 0000 0011
E) 0000 1100

Question 64

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-(S ? ?T) ? {[V ? (?S • T)] $\lor$ [V ? (?T • S)]}

A) 0011 1111
B) 0011 0000
C) 0101 1101
D) 1111 0101
E) 1111 1101

Question 65

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-{[X ? ?(Y $\lor$ Z)] • [Y ? ?(X $\lor$ Z)]} ? [(X • Z) $\lor$ ?(X • ?Z)]

A) 1111 1111
B) 1110 1111
C) 1010 1111
D) 0001 1110
E) 0001 0110

Question 66

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[(E ? ?F) $\lor$ (G ? ?H)] $\lor$ [(?E • ?F) • (?G • ?H)]

A) 1001 1111 1111 1001
B) 1001 1111 1111 1000
C) 0110 1111 1111 0110
D) 0110 1111 1111 0111
E) 0110 1111 1111 0000

Question 67

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the ? assignment; the second variable in the formula read left to right (if any) gets the ? assignment; the third variable in the formula read left to right (if any) gets the ? assignment; and the fourth variable in the formula read left to right (if any) gets the ? assignment.
For exercises with only one propositional variable, the standard assignment is:
$\begin{array} { |l| } \hline \alpha\\ \hline \text { 1}\\ \hline \text {0 }\\\hline\end{array}$

For exercises with two propositional variables, the standard assignment is:
$$\begin{array} { | c | c | } \hline \alpha& \boldsymbol { \beta } \\\hline 1 & 1 \\\hline 1 & 0 \\\hline 0 & 1 \\\hline0& 0\\\hline\end{array}$$
For exercises with three propositional variables, the standard assignment is:
$$\begin{array} { | c | c | c | } \hline \alpha & \beta & \gamma \\\hline 1 & 1 & 1 \\\hline 1 & 1 & 0 \\\hline 1 & 0 & 1 \\\hline 1 & 0& 0e \\\hline 0 & 1 & 1 \\\hline 0 & 1 & 0 \\\hline 0 &0& 1 \\\hline 0 & 0& 0 \\\hline\end{array}$$
For exercises with four propositional variables, the standard assignment is:
$$\begin{array} { | l | l | l | l | } \hline \alpha & \boldsymbol { \beta } &\gamma& \delta\ \\\hline 1 & 1 & 1 & 1 \\\hline 1 & 1 & 1 & 0 \\\hline 1 & 1 & 0 & 1 \\\hline 1 & 1 & 0 & 0 \\\hline 1 & 0 & 1 & 1 \\\hline 1 & 0 & 1 & 0\\\hline 1 & 0 & 0 & 1 \\\hline 1 & 0& 0 & 0 \\\hline 0 & 1 & 1 & 1 \\\hline 0 & 1 & 1 &0 \\\hline\end{array}$$
$$\begin{array} { | c | c | c | c | } \hline 0 & 1 & 0 & 1 \\\hline 0 & 1 & 0 & 0 \\\hline 0& 0 & 1 & 1 \\\hline 0 & 0 & 1 & 0 \\\hline 0 & 0 & 0 & 1 \\\hline 0 & 0 & 0 & 0 \\\hline\end{array}$$
$$\begin{array} { |l | l | l | l |} \hline 0 & 1 & 0 & 1 \\\hline\end{array}$$
For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions.

-[(A ? B) ? (?C ? D)] ? {[A • (B $\lor$ D)] ? [D • (B $\lor$ ?D)]}

A) 1111 1010 0000 0000
B) 1110 0001 1110 1110
C) 1011 1111 0101 1111
D) 1100 0100 1100 0100
E) 1110 1111 0011 1010

Question 68

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(I ⊃ ∼I) ⊃ [I ⊃ (I ⊃ ∼I)]

A) Tautology
B) Contingency
C) Contradiction

Question 69

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(G • ∼G) ⊃ (G $\lor$ ∼G)

A) Tautology
B) Contingency
C) Contradiction

Question 70

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(∼A • B) ≡ (B ⊃ A)

A) Tautology
B) Contingency
C) Contradiction

Question 71

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(A • ∼B) • (B $\lor$ ∼A)

A) Tautology
B) Contingency
C) Contradiction

Question 72

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(A ≡ ∼B) ⊃ ∼(B • A)

A) Tautology
B) Contingency
C) Contradiction

Question 73

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(C ⊃ ∼D) $\lor$ (∼D ⊃ C)

A) Tautology
B) Contingency
C) Contradiction

Question 74

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-(G ≡ H) ⊃ ∼[(G • H) $\lor$ (∼G • ∼H)]

A) Tautology
B) Contingency
C) Contradiction

Question 75

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-[N ⊃ (O ⊃ P)] ⊃ (N ⊃ O)

A) Tautology
B) Contingency
C) Contradiction

Question 76

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q $\lor$ S)

A) Tautology
B) Contingency
C) Contradiction

Question 77

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction.
-[(P $\lor$ Q) • (R • S)] ⊃ [(P • R) $\lor$ (Q • S)]

A) Tautology
B) Contingency
C) Contradiction

Question 78

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent.
-A ⊃ ∼A and ∼A ⊃ A

A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent

Question 79

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent.
-D • ∼E and ∼(E ⊃ ∼D)

A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent

Question 80

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent.
-G ≡ ∼H and (H • ∼G) $\lor$ (G • ∼H)

A) Logically equivalent
B) Contradictory
C) Neither logically equivalent nor contradictory, but consistent
D) Inconsistent