Question 1

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-The sum of one and two is three.

Accepted Answer

The answer of use the following translation key to write...

Question 2

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-The sum of one and the successor of one is divisible by three.

Accepted Answer

The answer of use the following translation key to write...

Question 3

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-If a number is divisible by three, then its successor is not.

Accepted Answer

The answer of use the following translation key to write...

Question 4

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-The successor of a number is greater than itself.

Accepted Answer

The answer of use the following translation key to write...

Question 5

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-No number is greater than the sum of that number and two.

Accepted Answer

The answer of use the following translation key to write...

Question 6

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-The sum of any number and two is equal to the sum of the successor of that number and one.

Accepted Answer

The answer of use the following translation key to write...

Question 7

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-If a number is divisible by three, then the sum of its successor with the successor of one is also divisible by three.

Accepted Answer

The answer of use the following translation key to write...

Question 8

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-Any number is divisible by two if, and only if, the successor of its successor is divisible by two.

Accepted Answer

The answer of use the following translation key to write...

Question 9

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-Numbers that are not divisible by three have sums with themselves that are also not divisible by three.

Accepted Answer

The answer of use the following translation key to write...

Question 10

use the following translation key to write the sentences below in FF. 
a: one
b: two 
c: three
f(x): the successor of x 
g(x,y): the sum of x and y 
Nx: x is a number
Dxy: x is divisible by y
Gxy: x is greater than y
-If the sum of the successors of any two numbers is divisible by three then the sum of two and the sum of those two numbers is divisible by three.

Accepted Answer

The answer of use the following translation key to write...

Question 11

1. (?x)(?y)f(x,y)=f(y,x)
2. (?x)f(x,o)=o / (?x)f(o,x)=o

Accepted Answer

The answer of 1. (?x)(?y)f(x,y)=f(y,x)
2. (?x)f(x,o)=o / (?x)f(o,x)=o...

Question 12

1. (?x)(?y)(Bxy ? Dyx)
2. (?x)Bxf(x) / (?x)Df(x)x

Accepted Answer

The answer of 1. (?x)(?y)(Bxy ? Dyx)
2. (?x)Bxf(x) / (?x)Df(x)x...

Question 13

1. (?x){Ax ? [Bx • Bf(x)]}
2. ?Bf(f(e)) / ?Af€

Accepted Answer

The answer of 1. (?x){Ax ? [Bx • Bf(x)]}
2. ?Bf(f(e))...

Question 14

1. (?x)(?y){Pf(x,y) ? [Pf(x,x) • Pf(y,y)]}
2. a=f(d,b)
3. Pa / Pf(b,b)

Accepted Answer

The answer of 1. (?x)(?y){Pf(x,y) ? [Pf(x,x) • Pf(y,y)]}
2. a=f(d,b)
3....

Question 15

1. (?x)(?y)[f(x)=y ? (Pax • Qay)]
2. ?Pab / ?(?x)f(b)=x

Accepted Answer

The answer of 1. (?x)(?y)[f(x)=y ? (Pax • Qay)]
2. ?Pab...

Question 16

1. (?x)(?y)(?z)[(Pxy • Pyz) ? Pxz]
2. (?x)Pxf(x) / (?x)Pxf(f(x))

Accepted Answer

The answer of 1. (?x)(?y)(?z)[(Pxy • Pyz) ? Pxz]
2. (?x)Pxf(x)...

Question 17

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)]
2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)]
 / (∃x)(∃y)[Pf(x,y) • Rxy • x≠y]

Accepted Answer

The answer of 1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)]
2. (∃x)[Px...

Question 18

1. (?x){Px ? (?y)[Py • f(x)=y]}
2. Pa • (?x)f(x)?a / (?x)[(?y)f(x)=y • ?(?y)f(y)=x]

Accepted Answer

The answer of 1. (?x){Px ? (?y)[Py • f(x)=y]}
2. Pa...

Question 19

1. (∀x)[(Px • Qx) ⊃ Rf(x)]
2. (∀x)[Rx ⊃ (∃y)Pxy]
3. ∼(∀x)(Px ⊃ ∼Qx) / (∃x)(∃y)Pxy

Accepted Answer

The answer of 1. (∀x)[(Px • Qx) ⊃ Rf(x)]
2. (∀x)[Rx...

Question 20

1. (∀x)(∀y)(∃z)Sf(x)yz
2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz $ \lor $ Mzyx)] / (∃x)(∃y)(∃z)∼Mzg(y)f(g(x))

Accepted Answer

The answer of 1. (∀x)(∀y)(∃z)Sf(x)yz
2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz \( \lor...

Quiz 5: Full First-Order Logic

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The sum of one and two is three.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The sum of one and the successor of one is divisible by three.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -If a number is divisible by three, then its successor is not.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The successor of a number is greater than itself.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -No number is greater than the sum of that number and two.

1. (?x)(?y)f(x,y)=f(y,x) 2. (?x)f(x,o)=o / (?x)f(o,x)=o

1. (?x)(?y)(Bxy ? Dyx) 2. (?x)Bxf(x) / (?x)Df(x)x

1. (?x){Ax ? [Bx • Bf(x)]} 2. ?Bf(f(e)) / ?Af€

1. (?x)(?y){Pf(x,y) ? [Pf(x,x) • Pf(y,y)]} 2. a=f(d,b) 3. Pa / Pf(b,b)

1. (?x)(?y)[f(x)=y ? (Pax • Qay)] 2. ?Pab / ?(?x)f(b)=x

1. (?x)(?y)(?z)[(Pxy • Pyz) ? Pxz] 2. (?x)Pxf(x) / (?x)Pxf(f(x))

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)] 2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)] / (∃x)(∃y)[Pf(x,y) • Rxy • x≠y]

1. (?x){Px ? (?y)[Py • f(x)=y]} 2. Pa • (?x)f(x)?a / (?x)[(?y)f(x)=y • ?(?y)f(y)=x]

1. (∀x)[(Px • Qx) ⊃ Rf(x)] 2. (∀x)[Rx ⊃ (∃y)Pxy] 3. ∼(∀x)(Px ⊃ ∼Qx) / (∃x)(∃y)Pxy

1. (∀x)(∀y)(∃z)Sf(x)yz 2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz $\lor$ Mzyx)] / (∃x)(∃y)(∃z)∼Mzg(y)f(g(x))

Quiz 5: Full First-Order Logic

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The sum of one and two is three.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The sum of one and the successor of one is divisible by three.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -If a number is divisible by three, then its successor is not.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -The successor of a number is greater than itself.

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -No number is greater than the sum of that number and two.

1. (?x)(?y)f(x,y)=f(y,x) 2. (?x)f(x,o)=o / (?x)f(o,x)=o

1. (?x)(?y)(Bxy ? Dyx) 2. (?x)Bxf(x) / (?x)Df(x)x

1. (?x){Ax ? [Bx • Bf(x)]} 2. ?Bf(f(e)) / ?Af€

1. (?x)(?y){Pf(x,y) ? [Pf(x,x) • Pf(y,y)]} 2. a=f(d,b) 3. Pa / Pf(b,b)

1. (?x)(?y)[f(x)=y ? (Pax • Qay)] 2. ?Pab / ?(?x)f(b)=x

1. (?x)(?y)(?z)[(Pxy • Pyz) ? Pxz] 2. (?x)Pxf(x) / (?x)Pxf(f(x))

1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)] 2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)] / (∃x)(∃y)[Pf(x,y) • Rxy • x≠y]

1. (?x){Px ? (?y)[Py • f(x)=y]} 2. Pa • (?x)f(x)?a / (?x)[(?y)f(x)=y • ?(?y)f(y)=x]

1. (∀x)[(Px • Qx) ⊃ Rf(x)] 2. (∀x)[Rx ⊃ (∃y)Pxy] 3. ∼(∀x)(Px ⊃ ∼Qx) / (∃x)(∃y)Pxy

1. (∀x)(∀y)(∃z)Sf(x)yz 2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz ∨ \lor ∨ Mzyx)] / (∃x)(∃y)(∃z)∼Mzg(y)f(g(x))

1. (∀x)(∀y)(∃z)Sf(x)yz 2. (∀x)(∀y)(∀z)[Sxyz ⊃ ∼(Cxyz $\lor$ Mzyx)] / (∃x)(∃y)(∃z)∼Mzg(y)f(g(x))