Transforming Statements Using Equivalence Laws Quiz

1) Which pair of statements is logically equivalent?

P → Q and Q → P

¬Q → ¬P and P → Q

P ∧ Q and P ∨ Q

¬(P ∧ Q) and P ∨ Q

The conditional statement P → Q is logically equivalent to its contrapositive ¬Q → ¬P.

Explanation

The conditional statement P → Q is logically equivalent to its contrapositive ¬Q → ¬P.

2) Which expression is logically equivalent to (P ∧ Q) ∨ (P ∧ R)?

P ∧ (Q ∨ R)

(P ∨ Q) ∧ (P ∨ R)

P ∨ (Q ∧ R)

(P ∧ Q) ∧ R

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

Explanation

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

3) ¬(P ∨ Q) is logically equivalent to:

¬P ∨ ¬Q

¬P ∧ ¬Q

P ∧ Q

¬(P ∧ Q)

This is De Morgan's Law: the negation of a disjunction is the conjunction of the negations.

Explanation

This is De Morgan's Law: the negation of a disjunction is the conjunction of the negations.

4) Which is logically equivalent to the statement 'It is not true that both Alice and Bob failed'?

Alice passed and Bob passed

Either Alice passed or Bob passed

Alice failed or Bob failed

Both Alice and Bob passed

The statement is equivalent to ¬(A ∧ B), where A is "Alice failed" and B is "Bob failed". By De Morgan's Law, this is equivalent to ¬A ∨ ¬B, which means either Alice passed or Bob passed.

Explanation

The statement is equivalent to ¬(A ∧ B), where A is "Alice failed" and B is "Bob failed". By De Morgan's Law, this is equivalent to ¬A ∨ ¬B, which means either Alice passed or Bob passed.

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

Q

P ∧ Q

P → Q

¬Q

We have the following chain of logical equivalences: (P → Q) ∧ P ≡ (¬P ∨ Q) ∧ P ≡ (¬P ∧ P) ∨ (Q ∧ P) ≡ F ∨ (Q ∧ P) ≡ P ∧ Q. This means if P is true and P implies Q, then Q must also be true.

Explanation

We have the following chain of logical equivalences: (P → Q) ∧ P ≡ (¬P ∨ Q) ∧ P ≡ (¬P ∧ P) ∨ (Q ∧ P) ≡ F ∨ (Q ∧ P) ≡ P ∧ Q. This means if P is true and P implies Q, then Q must also be true.

6) The statement P ∨ (P ∧ Q) is logically equivalent to:

P

Q

P ∧ Q

P ∨ Q

P ∨ (P ∧ Q) ≡ P. This is the absorption law. If P is true, then the whole expression is true regardless of Q.

Explanation

P ∨ (P ∧ Q) ≡ P. This is the absorption law. If P is true, then the whole expression is true regardless of Q.

7) Which statement is logically equivalent to 'You can't be both happy and sad'?

If you're happy, you're not sad

You're either happy or sad

If you're not happy, you're sad

You're not happy and not sad

The statement is equivalent to ¬(H ∧ S), where H is "happy" and S is "sad". By De Morgan's Law, this is equivalent to ¬H ∨ ¬S, which means if you're happy (H is true), then you're not sad (¬S is true).

Explanation

The statement is equivalent to ¬(H ∧ S), where H is "happy" and S is "sad". By De Morgan's Law, this is equivalent to ¬H ∨ ¬S, which means if you're happy (H is true), then you're not sad (¬S is true).

8) Which expression is logically equivalent to ¬P → ¬Q?

P → Q

Q → P

¬Q → ¬P

P ∧ Q

¬P → ¬Q is the contrapositive of Q → P. The contrapositive of any implication is logically equivalent to the original implication.

Explanation

¬P → ¬Q is the contrapositive of Q → P. The contrapositive of any implication is logically equivalent to the original implication.

9) Which pair is logically equivalent?

P→Q and ¬P∨Q

P→Q and P∧Q

P→Q and Q→P

P→Q and P∨Q

P → Q is equivalent to ¬P ∨ Q. This is the material implication equivalence.

Explanation

P → Q is equivalent to ¬P ∨ Q. This is the material implication equivalence.

10) ¬(P ↔ Q) is logically equivalent to:

P ↔ ¬Q

¬P ↔ Q

Exactly one of P or Q is true

All of the above

The negation of P ↔ Q (P if and only if Q) is equivalent to P ⊕ Q (P exclusive or Q), which means P and Q have different truth values. This is also equivalent to P ↔ ¬Q and ¬P ↔ Q.

Explanation

The negation of P ↔ Q (P if and only if Q) is equivalent to P ⊕ Q (P exclusive or Q), which means P and Q have different truth values. This is also equivalent to P ↔ ¬Q and ¬P ↔ Q.

11) Which statement is logically equivalent to ¬(P ∧ Q)?

¬P ∧ ¬Q

¬P ∨ ¬Q

P ∨ Q

P ∧ ¬Q

This is De Morgan's Law for the negation of AND. The negation of "P and Q" is equivalent to "not P OR not Q".

Explanation

This is De Morgan's Law for the negation of AND. The negation of "P and Q" is equivalent to "not P OR not Q".

12) P ∧ (Q ∨ ¬Q) is equivalent to:

P

Q ∨ ¬Q

Q ∧ ¬Q

¬P

Q ∨ ¬Q is always true (tautology). Therefore, P ∧ (Q ∨ ¬Q) is equivalent to P ∧ true, which is simply P.

Explanation

Q ∨ ¬Q is always true (tautology). Therefore, P ∧ (Q ∨ ¬Q) is equivalent to P ∧ true, which is simply P.

13) Which is logically equivalent to P ∨ (Q ∧ R)?

(P ∨ Q) ∧ (P ∨ R)

(P ∧ Q) ∨ R

P ∧ (Q ∨ R)

(P ∨ Q) ∧ R

This is the distributive Law of OR over AND: P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R).

Explanation

This is the distributive Law of OR over AND: P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R).

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

(P ∨ Q) → R

(P → Q) → R

P → (Q ∧ R)

(P ∧ Q) → R

By the exportation law, P → (Q → R) is logically equivalent to (P ∧ Q) → R.

Explanation

By the exportation law, P → (Q → R) is logically equivalent to (P ∧ Q) → R.

15) (P ∧ Q) ∧ R is equivalent to:

P ∧ (Q ∧ R)

P ∨ (Q ∨ R)

(P ∧ Q) ∨ R

P → (Q → R)

This is the associative law for conjunction. Since all the connectives are AND operations, the grouping of the AND operations does not affect the result.

Explanation

This is the associative law for conjunction. Since all the connectives are AND operations, the grouping of the AND operations does not affect the result.