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The contrapositive of the sentence \[ \sim p \to q\] is equivalent to:
A. $p \to \sim q$
B. $q \to \sim p$
C. $ \sim q \to p$
D. $ \sim p \to \sim q$
E. $ \sim q \to \sim p$

Last updated date: 22nd Jul 2024
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Given an if-then statement, then some related statements with their names are provided below:
StatementIf p, then q.p → q
InverseIf not p, then not q.$ \sim p \to \sim q$
ConverseIf q, then p.$q \to p$
ContrapositiveIf not q, then not p.$ \sim q \to \sim p$

1) In mathematical logic, $ \sim ( \sim p) = p$
2) A statement and its contrapositive are logically equivalent, in the sense that if the statement is true, then its contrapositive is true and vice versa.
3) Since an inverse is the contrapositive of the converse, inverse $ \sim p \to \sim q$ and converse $q \to p$ are also logically equivalent to each other.

Complete step by step solution:
From the definition, the contrapositive of $p \to q$ is $ \sim q \to \sim p$
Therefore, the contrapositive of $ \sim p \to q$ will be $ \sim q \to \sim ( \sim p)$
Since, $ \sim ( \sim p) = p$ , therefore $ \sim q \to \sim ( \sim p) = \sim q \to p$.
Therefore, the contrapositive of $ \sim p \to q$ is $ \sim q \to p$ .

The correct answer option is C.

Logical equivalence is different from material equivalence. Formulas for p and q are logically equivalent if and only if the statement of their material equivalence p ⇔ q is a tautology.
The material equivalence of p and q (often written as p ⇔ q) is itself another statement which expresses the idea “p if and only if q”.