Contrapositive Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Prove by contrapositive: 'If n2n^2 is odd, then nn is odd.'

Solution

  1. 1
    The contrapositive is: 'If nn is even, then n2n^2 is even.'
  2. 2
    Assume nn is even, so n=2kn = 2k for some integer kk.
  3. 3
    Then n2=(2k)2=4k2=2(2k2)n^2 = (2k)^2 = 4k^2 = 2(2k^2), which is even.
  4. 4
    Since the contrapositive is true, the original statement is true.

Answer

Proved: if n2 is odd, then n is odd.\text{Proved: if } n^2 \text{ is odd, then } n \text{ is odd.}
Proving the contrapositive is often easier than proving the original statement directly, especially when the negated hypothesis gives a concrete algebraic form to work with.

About Contrapositive

The contrapositive of a conditional statement P⇒QP \Rightarrow Q is ¬Q⇒¬P\neg Q \Rightarrow \neg P, formed by negating both parts and reversing their order — it is always logically equivalent to the original.

Learn more about Contrapositive →

More Contrapositive Examples

Example 1 easy

Form the contrapositive of: 'If [formula] is even, then [formula] is even.'

Example 3 easy

Write the contrapositive of: 'If a triangle is equilateral, then all its angles are [formula].'

Example 4 medium

Write the contrapositive of: 'If [formula] is even, then [formula] is even.'

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