Logical Equivalence With Conditionals And De Morgan’s Laws Quiz

1) Which of the following represents the logical equivalence of P → Q?

¬P ∨ Q

P ∧ Q

¬P ∧ Q

P ∨ Q

The implication P → Q means 'if P then Q'. This can be rephrased as 'either not P, or Q', written as ¬P ∨ Q. When P is false the implication is true; when P is true, the truth of the implication matches the truth of Q.

Explanation

The implication P → Q means 'if P then Q'. This can be rephrased as 'either not P, or Q', written as ¬P ∨ Q. When P is false the implication is true; when P is true, the truth of the implication matches the truth of Q.

2) The expression P ∧ (Q ∨ R) is equivalent to:

(P ∧ Q) ∨ (P ∧ R)

(P ∨ Q) ∧ (P ∨ R)

P ∨ (Q ∧ R)

P → (Q ∨ R)

This is the distributive law of logical AND over OR. P ∧ (Q ∨ R) means P is true AND either Q or R is true. This is equivalent to saying either (P ∧ Q) is true OR (P ∧ R) is true.

Explanation

This is the distributive law of logical AND over OR. P ∧ (Q ∨ R) means P is true AND either Q or R is true. This is equivalent to saying either (P ∧ Q) is true OR (P ∧ R) is true.

3) The expression ¬(P ∧ Q) is equivalent to:

¬P ∨ ¬Q

¬P ∧ ¬Q

P ∨ Q

P ∧ Q

This is De Morgan's Law. The negation of 'both P and Q' is equivalent to 'either not P or not Q'. If either P is false or Q is false, then P ∧ Q is false.

Explanation

This is De Morgan's Law. The negation of 'both P and Q' is equivalent to 'either not P or not Q'. If either P is false or Q is false, then P ∧ Q is false.

4) Which pair of statements is logically equivalent?

P ∧ Q and P ∨ Q

P → Q and Q → P

P → Q and ¬P ∨ Q

P ↔ Q and P → Q

P → Q is equivalent to ¬P ∨ Q. That means if P is false (¬P is true), then the whole statement is true regardless of Q. If P is true, then Q must also be true for the implication to hold. This is known as the material implication equivalence.

Explanation

P → Q is equivalent to ¬P ∨ Q. That means if P is false (¬P is true), then the whole statement is true regardless of Q. If P is true, then Q must also be true for the implication to hold. This is known as the material implication equivalence.

5) The negation of 'If P then Q' is:

¬P → ¬Q

P → ¬Q

P ∧ ¬Q

¬(P ∧ Q)

The negation of P → Q is equivalent to P ∧ ¬Q. This means 'P is true AND Q is false', which is the only case where 'if P then Q' is false.

Explanation

The negation of P → Q is equivalent to P ∧ ¬Q. This means 'P is true AND Q is false', which is the only case where 'if P then Q' is false.

6) The negation of 'P or Q' is equivalent to:

¬P ∨ ¬Q

¬P ∧ ¬Q

¬P → ¬Q

P ∧ Q

This is an application of De Morgan's Law. The negation of 'P or Q' is equivalent to 'not P AND not Q'. Both P and Q must be false for 'P or Q' to be false.

Explanation

This is an application of De Morgan's Law. The negation of 'P or Q' is equivalent to 'not P AND not Q'. Both P and Q must be false for 'P or Q' to be false.

7) (P ∧ Q) → R is logically equivalent to:

P → (Q → R)

(P → R) ∧ (Q → R)

(P ∨ Q) → R

P → (Q ∧ R)

We have the following chain of logical equivalences: (P ∧ Q) → R ≡ ¬(P ∧ Q) ∨ R ≡ ¬P ∨ ¬Q ∨ R ≡ ¬P ∨ (¬Q ∨ R) ≡ ¬P ∨ (Q → R) ≡ P → (Q → R). This is the exportation law: 'if P and Q then R' is equivalent to 'if P then (if Q then R)'.

Explanation

We have the following chain of logical equivalences: (P ∧ Q) → R ≡ ¬(P ∧ Q) ∨ R ≡ ¬P ∨ ¬Q ∨ R ≡ ¬P ∨ (¬Q ∨ R) ≡ ¬P ∨ (Q → R) ≡ P → (Q → R). This is the exportation law: 'if P and Q then R' is equivalent to 'if P then (if Q then R)'.

8) The teacher announced: 'If I study hard, I will pass.' Which of the following is the logically equivalent statement expressed solely as a disjunction (OR statement)?

If I don't pass, I didn't study hard'

If I pass, I studied hard'

I study hard or I won't pass'

I don't study hard or I will pass'

Using the material implication equivalence, 'If S then P' is equivalent to '¬S ∨ P', which translates to 'I don't study hard or I will pass'. Although 'If S then P' is also equivalent to 'If not P then not S', that form does not use a disjunction.

Explanation

Using the material implication equivalence, 'If S then P' is equivalent to '¬S ∨ P', which translates to 'I don't study hard or I will pass'. Although 'If S then P' is also equivalent to 'If not P then not S', that form does not use a disjunction.

9) Which of the following is the standard conditional form used to represent 'P unless Q' in introductory logic?

P ∨ Q

P → Q

¬Q → P

P ∧ Q

The meaning of 'P unless Q' is that Q represents the only circumstance that prevents P from being true. Therefore, if that circumstance Q does not occur (¬Q is true), then P must be true. Thus 'P unless Q' means 'if not Q, then P', which is ¬Q → P.

Explanation

The meaning of 'P unless Q' is that Q represents the only circumstance that prevents P from being true. Therefore, if that circumstance Q does not occur (¬Q is true), then P must be true. Thus 'P unless Q' means 'if not Q, then P', which is ¬Q → P.

10) Which of the following is a logical equivalence?

P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)

P → Q ≡ P ∧ Q

P ↔ Q ≡ P ∨ Q

¬(P ∨ Q) ≡ ¬P ∨ ¬Q

This is the distributive law of OR over AND. P ∨ (Q ∧ R) means P is true or both Q and R are true. This is equivalent to saying both (P ∨ Q) and (P ∨ R) are true.

Explanation

This is the distributive law of OR over AND. P ∨ (Q ∧ R) means P is true or both Q and R are true. This is equivalent to saying both (P ∨ Q) and (P ∨ R) are true.

11) P ∨ P is equivalent to:

P

¬P

P ∧ P

P ∨ ¬P

The idempotent law states that P ∨ P ≡ P. The expression 'P or P' is simply P.

Explanation

The idempotent law states that P ∨ P ≡ P. The expression 'P or P' is simply P.

12) The statement 'If it rains, I stay home' is logically equivalent to:

It rains or I stay home

If I don't stay home, it did rain

It doesn't rain or I stay home

I stay home and it rains

Using the material implication equivalence, 'If R then H' is equivalent to '¬R ∨ H', which translates to 'It doesn't rain or I stay home'.

Explanation

Using the material implication equivalence, 'If R then H' is equivalent to '¬R ∨ H', which translates to 'It doesn't rain or I stay home'.

13) P ↔ Q is logically equivalent to:

(P → Q) ∧ (Q → P)

(P → Q) ∨ (Q → P)

(P ∧ Q) ∨ (¬P ∧ ¬Q)

Both A and C

The biconditional statement P ↔ Q is equivalent to both (P → Q) ∧ (Q → P) and (P ∧ Q) ∨ (¬P ∧ ¬Q). The first form means 'P implies Q and Q implies P', and the second means 'both P and Q are true, or both P and Q are false'.

Explanation

The biconditional statement P ↔ Q is equivalent to both (P → Q) ∧ (Q → P) and (P ∧ Q) ∨ (¬P ∧ ¬Q). The first form means 'P implies Q and Q implies P', and the second means 'both P and Q are true, or both P and Q are false'.

14) Which expression is logically equivalent to (P → Q) ∧ (P → R)?

P → (Q ∧ R)

P → (Q ∨ R)

(P ∧ Q) → R

(Q → P) ∧ (R → P)

We can interpret (P → Q) ∧ (P → R) as the conjunction "if P is true, then Q is true" AND "if P is true, then R is true". This is logically equivalent to "if P is true, then both Q and R are true," which is expressed as P → (Q ∧ R). This follows from the distribution of implication over conjunction. If P being true requires both Q and R to be true, then P implies the conjunction of Q and R. The other options do not preserve the same truth conditions.

Explanation

We can interpret (P → Q) ∧ (P → R) as the conjunction "if P is true, then Q is true" AND "if P is true, then R is true". This is logically equivalent to "if P is true, then both Q and R are true," which is expressed as P → (Q ∧ R). This follows from the distribution of implication over conjunction. If P being true requires both Q and R to be true, then P implies the conjunction of Q and R. The other options do not preserve the same truth conditions.

15) The statement 'P if and only if Q' is logically equivalent to:

(P ∨ Q) ∨ (¬P ∨ ¬Q)

(P ∧ ¬Q) ∨ (¬P ∧ Q)

(P → Q) ∨ (Q → P)

(¬P ∧ ¬Q) ∨ (P ∧ Q)

The biconditional 'P if and only if Q' is true precisely when both propositions have the same truth value (both true or both false), which is expressed as (¬P ∧ ¬Q) ∨ (P ∧ Q).

Explanation

The biconditional 'P if and only if Q' is true precisely when both propositions have the same truth value (both true or both false), which is expressed as (¬P ∧ ¬Q) ∨ (P ∧ Q).