Lesson 1 — Conditional Statements
Big idea: A conditional statement is an if–then promise. The “if” part is the hypothesis; the “then” part is the conclusion.
Words to know
| Word | Meaning |
|---|---|
| Conditional statement | An if–then statement. |
| Hypothesis | The “if” part; symbol a. |
| Conclusion | The “then” part; symbol b. |
| Euler diagram | A picture of a conditional using a small circle inside a larger circle. |
Preview word: Converse — swap the hypothesis and conclusion. The converse of a → b is b → a.
Concepts to know
- A conditional is written a → b, read “If a, then b.”
- The hypothesis is the condition; the conclusion is what follows.
- In an Euler diagram, put a inside b:
- inside a → inside b;
- inside b does not guarantee inside a;
- outside b → outside a.
- “a if b” means b → a.
- “a only if b” means a → b.
- A true statement does not automatically make its converse true.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Identify the hypothesis | Find the “if” or “when” clause. |
| Identify the conclusion | Find the “then” clause. |
| Rewrite in if–then form | Put the condition first, then the result. |
| Draw an Euler diagram | Small circle = hypothesis; large circle = conclusion. |
| Read an Euler diagram | If you are in the small circle, you are in the large one. |
| Decide whether the converse is true | Test a case that satisfies the conclusion but not the hypothesis. |
| Interpret “only if” | “a only if b” means a → b, not b → a. |
Worked example
Statement: “If it is snowing, then it is cold outside.”
- Hypothesis: it is snowing.
- Conclusion: it is cold outside.
- Euler diagram: “snowing” inside “cold outside.”
- Converse: “If it is cold outside, then it is snowing.”
- The converse is false: it can be cold without snow.
Watch out
- Do not call the “then” part the hypothesis.
- Do not assume the converse is true because the original is true.
- “Only if” reverses the direction from “if.”
- An Euler diagram illustrates a statement; it does not prove it.