Contrapositive

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Logical equivalence: 'If A then B' is equivalent to 'If not B then not A'.

The Contrapositive is a logically equivalent form of a conditional statement. For any conditional 'If A then B', its contrapositive 'If not B then not A' has exactly the same truth value. This equivalence is fundamental in mathematics, particularly in proof by contrapositive, where proving the contrapositive is often easier than proving the original statement directly.

P → Q ≡ ¬Q → ¬P

• 'If it is a dog, then it is an animal' is equivalent to 'If it is not an animal, then it is not a dog'
• 'If n² is even, then n is even' is equivalent to 'If n is odd, then n² is odd'
• 'If you pass the exam, you studied' is equivalent to 'If you did not study, you did not pass the exam'

How to recognize it

• Start with an 'If A then B' statement
• Negate both parts: A becomes 'not A', B becomes 'not B'
• Swap their positions: 'If not B then not A'

Common mistakes

• Confusing contrapositive with converse: 'If B then A' is the converse, NOT equivalent to the original
• Confusing contrapositive with inverse: 'If not A then not B' is the inverse, NOT equivalent to the original
• Only the contrapositive preserves logical equivalence

Practice guidance

Logical Deduction