📐 Core Concept
The contrapositive of a conditional statement is formed by negating both the hypothesis and conclusion, then switching them. The contrapositive is always logically equivalent to the original.
Original: P → Q ≡ Contrapositive: ¬Q → ¬P
Converse: Q → P ≡ Inverse: ¬P → ¬Q (NOT equivalent to original)
| Name |
Form |
Equivalent to original? |
| Original |
P → Q |
✓ YES (itself) |
| Contrapositive |
¬Q → ¬P |
✓ YES (always) |
| Converse |
Q → P |
✗ NOT always |
| Inverse |
¬P → ¬Q |
✗ NOT always |
✏️ Worked Examples
Example 1 — Basic
Statement: "If it is raining, then the ground is wet."
P = "it is raining" | Q = "the ground is wet"
Contrapositive: "If the ground is not wet, then it is not raining." (¬Q → ¬P) ✓
Converse: "If the ground is wet, then it is raining." (not equivalent)
Inverse: "If it is not raining, then the ground is not wet." (not equivalent)
Example 2 — Math Context
Statement: "If n is even, then n² is even."
Contrapositive: "If n² is not even (odd), then n is not even (odd)." ✓
This contrapositive is also true — and can be used to prove the original!
Example 3 — Identifying Equivalence
Which is equivalent to "If P then Q"?
(A) If Q then P → Converse — NOT equivalent
(B) If not P then not Q → Inverse — NOT equivalent
(C) If not Q then not P → Contrapositive — EQUIVALENT ✓
Answer: (C)
Example 4 — Proof by Contrapositive
Prove: If n² is odd, then n is odd. (for integers)
Write contrapositive: "If n is not odd (i.e., n is even), then n² is not odd (i.e., n² is even)."
Proof: If n = 2k, then n² = 4k² = 2(2k²), which is even. ✓
Since the contrapositive is true, the original statement is true.