Prove P → Q by proving the equivalent ¬Q → ¬P
“If P then Q” says exactly the same thing as “if not Q then not P.” Sometimes the second version is far easier to prove.
Pick a statement, press Show Contrapositive, and compare the truth tables row by row. Both are false only when P is true and Q is false.
See why P → Q and ¬Q → ¬P are logically equivalent. They have the same truth values in every possible scenario!
| P | Q | P → Q | ¬Q | ¬P | ¬Q → ¬P | Match? |
|---|---|---|---|---|---|---|
| T | T | T | F | F | T | ✓ |
| T | F | F | T | F | F | ✓ |
| F | T | T | F | T | T | ✓ |
| F | F | T | T | T | T | ✓ |
Notice: P → Q and ¬Q → ¬P have identical truth values in every row!
P → Q ≡ ¬Q → ¬P. These are not just "related"—they're the exact same statement in different logical clothing. Proving one automatically proves the other. This is why contrapositive is so powerful: you're not taking a detour, you're walking the same path in reverse!
Proof by contrapositive exploits a fundamental fact of logic: the statement P → Q is logically equivalent to ¬Q → ¬P. Instead of proving P → Q directly, we prove ¬Q → ¬P. This is not a roundabout method -- it is proving the exact same thing in a different form.
Use contrapositive when the negation of Q gives you something concrete to work with, or when the direct path from P to Q feels unnatural.
See why P → Q and ¬Q → ¬P have identical truth tables. They are false in exactly the same situation: when P is true and Q is false.
Key insight: Do not confuse the contrapositive (¬Q → ¬P) with the converse (Q → P). The converse is a completely different statement and is NOT equivalent to the original.
Read each statement and choose the best technique, then compare how contrapositive and contradiction flow side by side.
These two techniques are often confused! Let's clarify the differences and practice choosing the right approach.
| Aspect | Contrapositive | Contradiction |
|---|---|---|
| What you prove | An equivalent statement (¬Q → ¬P) | The original statement (P) by showing ¬P is impossible |
| What you assume | ¬Q (the negation of the conclusion) | ¬P (the negation of what you want to prove) |
| What you derive | ¬P (the negation of the hypothesis) | A logical impossibility (X ∧ ¬X) |
| Type of proof | Direct proof of an equivalent statement | Indirect proof via impossibility |
| Best used when | ¬Q gives useful concrete information | Direct/contrapositive seem hard, or proving existence/uniqueness |
Prove: "If n² is odd, then n is odd"
What's the best proof technique?
These two techniques are often confused. Contrapositive proves an equivalent statement directly; contradiction seeks an impossibility. See the difference side-by-side.
Key insight: A contrapositive proof is really a direct proof of an equivalent statement. It never needs to reach a contradiction -- it just proves ¬Q → ¬P in a straight line.
Pick a statement and press Next Step to walk through its contrapositive proof, such as “if n is odd, then n² is odd.”
Walk through classic contrapositive proofs step by step. See how assuming ¬Q leads us to prove ¬P.
We want to prove: If n² is even, then n is even.
Walk through the classic example: "if n² is even, then n is even." The direct proof is awkward, but the contrapositive -- "if n is odd, then n² is odd" -- is a clean two-line calculation.
Key insight: Contrapositive shines for statements about divisibility and parity. "If n is odd" immediately gives n = 2k+1, which is easy to square and analyze.
Spot a mistake or have an idea?