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Core explanation

A proof by contradiction assumes the statement you want to prove is false, and then shows this assumption leads to something impossible.

General structure:

Assume the statement is false.
Derive logical consequences of that assumption.
Reach a contradiction, such as:
- A number is both even and odd.
- 0 = 1.
- A positive number is < 0.
Conclude the original statement must be true.

Contradiction is powerful when:

The negation of the claim directly violates a known theorem.
The structure of the problem naturally leads to two mutually exclusive conclusions.
The goal is easier to prove impossible than true.

Example: Integers

Prove: There is no smallest positive rational number.

Assume there is a smallest positive rational number r.
Then r/2 is also positive and rational, but r/2 < r — contradiction.

Thus no such smallest rational exists.

Contradiction proofs are fundamental in analysis, algebra, set theory, and number theory.

TL;DR — key idea

In contradiction proofs, you assume the opposite of what you want to prove and show that assumption leads to an impossibility.

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Use proof by contradiction to prove ONE: 1. There is no largest odd integer. 2. If a² is even, then a is even. 3. Between any two rational numbers, there is another rational number. Explain exactly where the contradiction occurs.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.