139k views
4 votes
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

1 Answer

0 votes

Final answer:

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.
User Scymex
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories