Question

Let p and q be two propositions. Then the contrapositive the implication \[p \to q\]
A. \[ \sim q \to \sim p\]
B. \[ \sim p \to \sim q\]
C. \[q \to p\]
D. \[p \leftrightarrow q\]

Answer 1

Hint: The contrapositive of an implication is an implication with the antecedent and consequent negated and interchanged, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.

Complete step by step solution:
The contrapositive is a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition or theorem and interchanging them as:
The contrapositive of \[p \to q\] is \[\neg q \to \neg p\]
The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.
The contrapositive of a conditional statement of the form "If p then q" is " if \[ \sim q\] then \[ \sim p\]". Symbolically, the contrapositive of p q is \[ \sim q \sim p\]. A conditional statement is logically equivalent to its contrapositive.
Therefore, the contrapositive of \[p \to q\] is \[ \sim q \to \sim p\].
Hence, option A is the right answer.
So, the correct answer is “Option A”.

Note: Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here i.e., \[\neg p \to \neg q\].
Note that if \[p \to q\] is true and one is given that Q is false i.e., \[\neg q\], then it can logically be concluded that P must be also false i.e., \[\neg p\]. This is often called the law of contrapositive, or the modus tollens rule of inference and also note that an implication and its contrapositive are logically equivalent. To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.