GATE Overflow | Mathematics | Test 1 | Question: 29

Question

According to the principle of logic, an implication and it's contrapositive must be

• A. both true or both false
• B. both true
• C. both false
• D. none
   

Answer 1

Direct and Contrapositive always have same truth value.
Inverse and Converse always have same truth value.

Answer 2

Implication and contrapositive both can same values in truth table and at same time they can be either true of false So

option (A) is correct .

Answer 3

1. $P\rightarrow Q$ // F
    
2. $Q\rightarrow P$ //Converse of F
    
3. $\sim P\rightarrow \sim Q$ // Inverse of F
    
4. $\sim Q\rightarrow \sim P$ // Contrapositive of F

 

F is actually equivalent to contrapositive of F. So both have the same value. Option A

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If you take the converse of F, and apply contraposition to it, you'll find that converse of F is actually equivalent to the inverse of F. They have the same value too.

In other words: 1 is equivalent to 4; 2 is equivalent to 3.