Lesson 4 — Indirect Proof
Big idea: To prove a statement indirectly, assume its opposite. If that assumption leads to a contradiction, the original statement must be true.
Words to know
| Word | Meaning |
|---|---|
| Indirect proof | A proof by contradiction: assume the opposite, then derive a contradiction. |
| Contradiction | A statement that cannot be true. |
Concepts to know
- The pattern is:
To prove: b
Suppose: not b
Then: ... → a contradiction
Therefore: not b is false, so b is true.- In a direct proof, you start with the hypothesis and chain to the conclusion.
- In an indirect proof, you start with the opposite of the conclusion and chain to a contradiction.
- The contradiction shows the assumption was false.
- Then the original conclusion must be true.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Write the first assumption | Assume the opposite of the conclusion. |
| Fill in missing statements | Chain from the assumption until something impossible happens. |
| Identify the contradiction | Look for a statement that conflicts with a given fact, definition, postulate, or theorem. |
| Write the final conclusion | State that the assumption was false, so the original statement is true. |
| Solve a logic puzzle | Test the opposite; eliminate possibilities that contradict the facts. |
| Prove “cannot” or “not” | Indirect proof is often the easiest choice. |
Worked example
Claim: Two lines cannot intersect in more than one point.
- Suppose two different lines intersect in two points.
- Then those two points lie on both lines.
- But two points determine exactly one line.
- So the two “different” lines would have to be the same line — a contradiction.
- Therefore, two lines cannot intersect in more than one point.
The hat example
Three hats: two red, one white. Tom wears red and does not know his own color.
Claim: Your hat is red.
- Suppose your hat is not red.
- Then it is white.
- If your hat is white, Tom would know his hat is red.
- But Tom does not know. Contradiction.
- Therefore, your hat is red.
Watch out
- Assume the opposite of the conclusion, not the conclusion.
- A surprising result is not automatically a contradiction.
- The contradiction must conflict with an accepted fact.
- Always state the final conclusion after the contradiction.