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.
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
Worked example
- To show
√2is irrational, assume√2 = a/bin lowest terms. - Then
a^2 = 2b^2, soais even; writea=2k. - Substituting back forces
beven 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 it
- 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.
Next recommended lesson
Continue through this topic with Least You Need to Know: Direct Proof and Contrapositive.
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