Logical Equivalence with Conditionals and De Morgan’s Laws Quiz

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1) Which of the following represents the logical equivalence of P → 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.

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About This Quiz
Logical Equivalence With Conditionals And De Morgans Laws Quiz - Quiz

Are you ready to see how different-looking statements can actually mean the same thing? In this quiz, you’ll work with conditionals, their disjunctive forms, De Morgan’s laws, and biconditionals to rewrite and analyze logical expressions. You’ll practice turning “if…then” statements into OR statements, negating implications, and understanding phrases like “unless”... see morein precise logical form. By the end, you’ll be more confident spotting when two statements are logically equivalent, even if they look completely different on the surface. see less

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

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.

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

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.

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4) Which pair of statements is logically equivalent?

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.

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

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.

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

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.

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

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)'.

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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)?

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.

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

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.

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10) Which of the following is a logical equivalence?

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.

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11) P ∨ P is equivalent to:

Explanation

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

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

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'.

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13) P ↔ Q is logically equivalent to:

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'.

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

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

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).

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Which of the following represents the logical equivalence of P → Q?
The expression P ∧ (Q ∨ R) is equivalent to:
The expression ¬(P ∧ Q) is equivalent to:
Which pair of statements is logically equivalent?
The negation of 'If P then Q' is:
The negation of 'P or Q' is equivalent to:
(P ∧ Q) → R is logically equivalent to:
The teacher announced: 'If I study hard, I will pass.' Which of the...
Which of the following is the standard conditional form used to...
Which of the following is a logical equivalence?
P ∨ P is equivalent to:
The statement 'If it rains, I stay home' is logically equivalent to:
P ↔ Q is logically equivalent to:
Which expression is logically equivalent to (P → Q) ∧ (P → R)?
The statement 'P if and only if Q' is logically equivalent to:
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