Question

For a conditional statement p → q, write down its inverse, its converse, and its...

a) Contrapositive, biconditional, hypothesis
b) Converse, contrapositive, biconditional
c) Hypothesis, biconditional, converse
d) Biconditional, contrapositive, hypothesis

Answer 1

In a conditional statement p → q, the converse is q → p, the inverse is ¬p → ¬q, the contrapositive is ¬q → ¬p, and the biconditional is p ↔ q. The hypothesis is p. Hence, the correct answer from the options provided is d) Biconditional, contrapositive, hypothesis.

Step-by-step explanation:

For a conditional statement p → q, we can identify several related logical statements. Here are the definitions and examples of each:

• Converse: The converse of a conditional statement swaps the hypothesis and conclusion. In symbols, the converse of p → q is q → p.
• Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. In symbols, the inverse is ¬p → ¬q (¬ represents negation).
• Contrapositive: The contrapositive negates and swaps the hypothesis and conclusion of the original statement. In symbols, the contrapositive of p → q is ¬q → ¬p.
• Biconditional: The biconditional combines the original statement and its converse, indicating that both are true. In symbols, the biconditional is p ↔ q, read as 'p if and only if q'.
• Hypothesis: This is the antecedent or the 'if' part of the conditional statement. In the statement p → q, p is the hypothesis.

To answer the student's question with the correct terminology and symbols:

• The inverse of p → q is ¬p → ¬q.
• The converse of p → q is q → p.
• The contrapositive of p → q is ¬q → ¬p.
• The biconditional is p ↔ q.
• The hypothesis is p.
• Therefore, the correct answer is d) Biconditional, contrapositive, hypothesis.