Math Problem Statement

Verify that the conditional statement PQ is equivalent to its converse, inverse and contra-positive forms.

Solution

To verify whether the conditional statement PQP \rightarrow Q is equivalent to its converse, inverse, and contrapositive, let us examine each form in detail.

Definitions:

  1. Original statement (conditional):
    PQP \rightarrow Q — "If PP, then QQ."

  2. Converse:
    QPQ \rightarrow P — "If QQ, then PP."

  3. Inverse:
    ¬P¬Q\neg P \rightarrow \neg Q — "If not PP, then not QQ."

  4. Contrapositive:
    ¬Q¬P\neg Q \rightarrow \neg P — "If not QQ, then not PP."

Truth Table Approach:

We can create a truth table to examine whether these forms are logically equivalent.

PPQQPQP \rightarrow QQPQ \rightarrow P¬P\neg P¬Q\neg Q¬P¬Q\neg P \rightarrow \neg Q¬Q¬P\neg Q \rightarrow \neg P
TTTTFFTT
TFFTFTTF
FTTFTFFT
FFTTTTTT

Analysis of the Truth Table:

  1. Original (P → Q): True except when PP is true and QQ is false.
  2. Converse (Q → P): True when both are true or both are false, but false when PP is false and QQ is true.
  3. Inverse (¬P → ¬Q): Same truth values as the converse.
  4. Contrapositive (¬Q → ¬P): Matches the truth values of the original conditional.

Conclusion:

  • The original statement PQP \rightarrow Q is logically equivalent to its contrapositive ¬Q¬P\neg Q \rightarrow \neg P.
  • The converse QPQ \rightarrow P is not equivalent to the original statement.
  • The inverse ¬P¬Q\neg P \rightarrow \neg Q is also not equivalent to the original statement, but it is equivalent to the converse.

Equivalences:

  • PQ¬Q¬PP \rightarrow Q \equiv \neg Q \rightarrow \neg P (Original and Contrapositive)
  • QP¬P¬QQ \rightarrow P \equiv \neg P \rightarrow \neg Q (Converse and Inverse)

Would you like more details on why these equivalences hold, or have any further questions?

Here are 5 related questions to explore further:

  1. What is the difference between logical equivalence and implication?
  2. How do you construct the truth table for a biconditional statement?
  3. What are tautologies and how do they relate to logical equivalence?
  4. How can you apply contrapositive reasoning in mathematical proofs?
  5. What is the significance of the converse in geometry, especially in theorems?

Tip: When proving logical equivalence, always rely on truth tables or known equivalence laws (such as De Morgan's laws) to verify correctness.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logic
Conditional Statements
Logical Equivalence

Formulas

P → Q
Converse: Q → P
Inverse: ¬P → ¬Q
Contrapositive: ¬Q → ¬P

Theorems

Truth Table Method for Logical Equivalence
Equivalence of Conditional and Contrapositive

Suitable Grade Level

Grades 10-12