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 is equivalent to its converse, inverse, and contrapositive, let us examine each form in detail.
Definitions:
-
Original statement (conditional):
— "If , then ." -
Converse:
— "If , then ." -
Inverse:
— "If not , then not ." -
Contrapositive:
— "If not , then not ."
Truth Table Approach:
We can create a truth table to examine whether these forms are logically equivalent.
T | T | T | T | F | F | T | T |
T | F | F | T | F | T | T | F |
F | T | T | F | T | F | F | T |
F | F | T | T | T | T | T | T |
Analysis of the Truth Table:
- Original (P → Q): True except when is true and is false.
- Converse (Q → P): True when both are true or both are false, but false when is false and is true.
- Inverse (¬P → ¬Q): Same truth values as the converse.
- Contrapositive (¬Q → ¬P): Matches the truth values of the original conditional.
Conclusion:
- The original statement is logically equivalent to its contrapositive .
- The converse is not equivalent to the original statement.
- The inverse is also not equivalent to the original statement, but it is equivalent to the converse.
Equivalences:
- (Original and Contrapositive)
- (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:
- What is the difference between logical equivalence and implication?
- How do you construct the truth table for a biconditional statement?
- What are tautologies and how do they relate to logical equivalence?
- How can you apply contrapositive reasoning in mathematical proofs?
- 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.
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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
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