Proof / Contradiction Patterns
Least You Need to Know: Contradiction Patterns
In proof by contradiction, assume the target claim is false and drive the assumption to something impossible, often a parity clash or a definition failure.
The least you need to know
- A contradiction proof starts by assuming the negation of the conclusion.
- The contradiction must conflict with a known fact or definition.
- Parity arguments often end with a number being both even and odd.
- You still need a logical chain, not just the word 'contradiction'.
Key notation
Assume ¬P
start contradiction by negating the target
2k
an even integer
2k+1
an odd integer
Tiny worked example
- To show `√2` is irrational, assume `√2 = a/b` in lowest terms.
- Then `a^2 = 2b^2`, so `a` is even; write `a=2k`.
- Substituting back forces `b` even too, contradicting lowest terms.
Common mistakes
- Students often assume the original claim instead of its negation.
- Students often stop at a surprising statement without showing why it is impossible.
- Students often confuse contradiction with contrapositive.
How to recognize this kind of problem
- Name the assumption clearly before manipulating it.
- If the contradiction uses parity, rewrite even numbers as `2k`.
- Ask exactly which definition or fact is being violated.