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Given the conditional statement ~p -> q which statement is logically equivalent?​

User James Waddington
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1 Answer

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Any implication is logically equivalent to its contrapositive. In other words,

¬pq¬qp

(¬ means the same thing as ~, "not")

To prove this: recall that

pq ⇔ ¬pq

This is because pq is true if p is false, or both p and q are true, i.e.

pq ⇔ ¬p ∨ (pq)

Disjunction (∨ or "or") distributes over conjunction (∧ or "and"), so that

pq ⇔ (¬pp) ∧ (¬pq)

but ¬pp is always true, or a tautology, so we're just left with ¬pq.

Then

¬pqpq

… ⇔ qp

… ⇔ ¬(¬q) ∨ p

… ⇔ ¬qp

User Navneeth
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