Lesson 3 — Direct Proof
Big idea: A direct proof is a chain of if–then statements that starts with the hypothesis and ends with the conclusion.
Words to know
| Word | Meaning |
|---|---|
| Syllogism | An argument of the form a → b, b → c, therefore a → c. |
| Direct proof | A proof that chains forward from the hypothesis to the conclusion. |
| Premises | The starting statements of the argument. |
| Conclusion | The statement reached at the end. |
| Theorem | A statement proved by reasoning from accepted statements. |
Concepts to know
- The basic pattern is:
a → b
b → c
Therefore, a → c- Each conclusion becomes the next hypothesis.
- A direct proof can have many links, not just two.
- If all premises are true and the logic is valid, the conclusion must be true.
- If a premise is false, the proof fails; the conclusion may still be true, but this argument has not proved it.
- A theorem is the conclusion of a valid proof.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Arrange statements in logical order | Put a statement whose conclusion matches the next statement’s hypothesis. |
| Write the missing statement | Match the “then” of one line to the “if” of the next. |
| Write the conclusion | Link the first hypothesis to the final conclusion. |
| Identify the premises | The premises are the starting if–then statements. |
| Identify the theorem | The theorem is the final conditional statement. |
| Use symbols | Replace each clause with a letter and write a → b, b → c, etc. |
Worked example
If you live at the South Pole, then you live in the Antarctic.
If you live in the Antarctic, then you live where it is cold.
If you live where it is cold, then you see a lot of penguins.
Therefore, if you live at the South Pole, you see a lot of penguins.The hypothesis of each line matches the conclusion of the line before it.
Watch out
- Do not link statements whose parts do not match.
- Do not reverse a conditional in the middle of the chain.
- A false premise does not necessarily make the conclusion false; it makes the proof invalid.
- Do not skip the final “therefore” statement.