What is the contrapositive of 'If a number is divisible by 6, then it is divisible by 3'?
AIf a number is divisible by 3, then it is divisible by 6
BIf a number is not divisible by 6, then it is not divisible by 3
CIf a number is not divisible by 3, then it is not divisible by 6
DIf a number is divisible by 3, then it is not divisible by 6
The contrapositive of P → Q is ¬Q → ¬P: negate both parts and swap their positions. P = 'divisible by 6,' Q = 'divisible by 3.' Contrapositive: 'If not divisible by 3, then not divisible by 6.' This is logically equivalent to the original — and also clearly true (if 3 does not divide a number, 6 certainly cannot). Option A is the converse; option B is the inverse.
Question 2 True / False
A conditional statement and its converse are generally logically equivalent.
TTrue
FFalse
Answer: False
A conditional and its converse can have different truth values. 'If it is a square, then it has four sides' is true, but its converse 'If it has four sides, then it is a square' is false (rectangles have four sides but are not squares). Only the contrapositive is guaranteed to match the original.
Question 3 Short Answer
Given the statement 'If an animal is a penguin, then it cannot fly,' write the converse, inverse, and contrapositive, and state which are logically equivalent to the original.
Think about your answer, then reveal below.
Model answer: Converse: 'If an animal cannot fly, then it is a penguin.' Inverse: 'If an animal is not a penguin, then it can fly.' Contrapositive: 'If an animal can fly, then it is not a penguin.' The contrapositive is logically equivalent to the original. The converse and inverse are equivalent to each other but not to the original.
The converse is false (ostriches cannot fly but are not penguins). The inverse is false (ostriches are not penguins but cannot fly). The contrapositive is true — any animal that can fly is definitely not a penguin. This confirms the pattern: original ↔ contrapositive, and converse ↔ inverse.