Question 1

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$p \wedge ( q \vee p )$

Accepted Answer

The compound statement $p \wedge ( q \vee p )$ simplifies to $p \wedge True$ because $q \vee p$ is True (since $p$ is True). Therefore, the entire statement is True since $p$ is True.

Question 2

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols.
-Jim does not play football or Michael plays basketball.

Accepted Answer

The statement "Jim does not play football or Michael plays basketball" can be symbolized as $\sim p \vee q$, where $\sim p$ represents "Jim does not play football" and $q$ represents "Michael plays basketball".

Question 3

Decide whether the statement is true or false.
-At least one irrational number is not an integer.

Accepted Answer

Irrational numbers, by definition, cannot be expressed as a fraction of two integers, which means they cannot be integers themselves.

Question 4

Decide whether the statement is true or false.
-Some real numbers are integers.

Accepted Answer

This statement is true. Integers are a subset of real numbers, so there are some real numbers that are also integers.

Question 5

Decide whether the statement is true or false.
-The absolute value of any number is positive.

Accepted Answer

The answer of Decide whether the statement is true or...

Question 6

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols.
-Jim does not play football or Michael does not play basketball.

Accepted Answer

The statement "Jim does not play football or Michael does not play basketball" translates to $\sim p \vee \sim q$, where $\sim$ represents negation and $\vee$ represents the logical OR operation.

Question 7

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$[ ( \sim p \wedge \sim q ) \vee \sim q ]$

Accepted Answer

The given compound statement simplifies as follows: Since $p$ is true, $\sim p$ (not $p$) is false. Since $q$ is false, $\sim q$ (not $q$) is true. Therefore, $(\sim p \wedge \sim q)$ simplifies to $(\text{false} \wedge \text{true})$, which is false. However, the compound statement also includes $\sim q$ (which is true) as part of an OR operation $(\vee)$. Since an OR operation is true if at least one of the components is true, the entire compound statement $[(\sim p \wedge \sim q) \vee \sim q]$ is true because $\sim q$ is true.

Question 8

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols.
-It is not the case that Jim does not play football and Michael does not play basketball.

Accepted Answer

The statement "It is not the case that Jim does not play football and Michael does not play basketball" translates to the negation of both statements being false together, which is $\sim ( \sim p \wedge \sim q )$.

Question 9

Decide whether the statement is true or false.
-Every rational number is an integer.

Accepted Answer

Rational numbers can be written as fractions, in which the numerator and denominator can be integers. However, not all rational numbers are integers. For example, 1/2 is a rational number, but not an integer. Integers are whole numbers, both positive and negative, without any fraction or decimal parts.

Question 10

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$\mathrm { p } \vee \sim \mathrm { q }$

Accepted Answer

Since $q$ is false, $\sim q$ (the negation of $q$) is true. The statement $p \vee \sim q$ is a disjunction, meaning it is true if at least one of the statements it connects is true. Since both $p$ and $\sim q$ are true, the compound statement is true.

Question 11

Decide whether the statement is true or false.
-Not every whole number is a real number.

Accepted Answer

Every whole number is a real number because the set of real numbers includes all the rational and irrational numbers, and whole numbers are part of the rational numbers.

Question 12

Decide whether the statement is true or false.
-Not every real number is not a whole number.

Accepted Answer

Some real numbers are whole numbers, so the statement that not every real number is not a whole number is true.

Question 13

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols.
-Jim plays football and Michael plays basketball.

Accepted Answer

The statement "Jim plays football and Michael plays basketball" is a conjunction of p and q, represented symbolically as $p \wedge q$.

Question 14

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$\sim \mathrm { p } \vee \mathrm { q }$

Accepted Answer

Since p is true and q is false, the statement $\sim \mathrm { p } \vee \mathrm { q }$ is true because the disjunction of a true statement and a false statement is always true.

Question 15

Decide whether the statement is true or false.
-No rational number is not a whole number.

Accepted Answer

Rational numbers include fractions and decimals that can be expressed as a ratio of two integers, not all of which are whole numbers.

Question 16

Decide whether the statement is true or false.
-Some whole numbers are not integers.

Accepted Answer

Whole numbers (0, 1, 2, 3, ...) are a subset of integers, which also include negative numbers. Therefore, all whole numbers are integers.

Question 17

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$\sim ( p \vee \sim q )$

Accepted Answer

The negation of a disjunction is true if and only if both disjuncts are false. Since q is false, $\sim q$ is true. Since p is true, $p \vee \sim q$ is true. Therefore, $\sim ( p \vee \sim q )$ is false.

Question 18

Decide whether the statement is true or false.
-There exists a rational number that is an integer.

Accepted Answer

Rational numbers are defined as numbers that can be expressed as a fraction or ratio of two integers, where the denominator is not zero. Since any integer can be expressed as a fraction by having the denominator as 1 (for example, 5 can be written as 5/1), it follows that integers are a subset of rational numbers.

Question 19

Decide whether the statement is true or false.
-All whole numbers are real numbers.

Accepted Answer

Whole numbers (0, 1, 2, 3, ...) are a subset of the real numbers, which include all the numbers that can be found on the number line, including both rational and irrational numbers.

Question 20

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement.
-$p \wedge q$

Accepted Answer

The answer of Let p represent a true statement and...

Quiz 3: Introduction to Logic

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $p \wedge ( q \vee p )$

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols. -Jim does not play football or Michael plays basketball.

Decide whether the statement is true or false. -At least one irrational number is not an integer.

Decide whether the statement is true or false. -Some real numbers are integers.

Decide whether the statement is true or false. -The absolute value of any number is positive.

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols. -Jim does not play football or Michael does not play basketball.

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $[ ( \sim p \wedge \sim q ) \vee \sim q ]$

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols. -It is not the case that Jim does not play football and Michael does not play basketball.

Decide whether the statement is true or false. -Every rational number is an integer.

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $\mathrm { p } \vee \sim \mathrm { q }$

Decide whether the statement is true or false. -Not every whole number is a real number.

Decide whether the statement is true or false. -Not every real number is not a whole number.

Let p represent the statement, "Jim plays football", and let q represent the statement "Michael plays basketball". Convert the compound statement into symbols. -Jim plays football and Michael plays basketball.

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $\sim \mathrm { p } \vee \mathrm { q }$

Decide whether the statement is true or false. -No rational number is not a whole number.

Decide whether the statement is true or false. -Some whole numbers are not integers.

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $\sim ( p \vee \sim q )$

Decide whether the statement is true or false. -There exists a rational number that is an integer.

Decide whether the statement is true or false. -All whole numbers are real numbers.

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. - $p \wedge q$