Conditional Statements and Logic
Write and analyze conditional (if-then) statements. Form and evaluate the converse, inverse, and contrapositive. Understand why the contrapositive is logically equivalent to the original and apply to geometric proof.
A lesson of Geometry, in Foundations of Geometry, about 10 minutes.
By the end of this lesson, you will be able to write conditional statements in if-then form, identify hypothesis and conclusion, form the converse/inverse/contrapositive, and determine the truth value of each related conditional.
The Big Idea
A conditional statement is a logical claim of the form "If $p$, then $q$" — the foundation of all mathematical reasoning. Understanding how to manipulate these statements reveals why some "flips" of a true statement are also true (the contrapositive) while others can be false (the converse) — a subtlety that trips up many students and researchers alike.
Structure of a Conditional
Conditional: "If $p$, then $q$" → written $p \to q$
- Hypothesis ($p$): the "if" part — the condition
- Conclusion ($q$): the "then" part — what follows
Example: "If a figure is a square, then it has four right angles."
- Hypothesis: "a figure is a square"
- Conclusion: "it has four right angles"
Related Conditionals
Given the conditional $p \to q$:
| Form | Statement | Relation to original |
|---|---|---|
| Conditional | $p \to q$ | Original |
| Converse | $q \to p$ | Swap hypothesis and conclusion |
| Inverse | $\neg p \to \neg q$ | Negate both |
| Contrapositive | $\neg q \to \neg p$ | Swap AND negate both |
Key truth relationship:
- A conditional and its contrapositive always have the same truth value.
- A conditional and its converse do NOT necessarily have the same truth value.
- The inverse and the converse always have the same truth value (both equivalent to each other but not necessarily to the original).
Example 1
Conditional: "If it is raining, then I will bring an umbrella." (Assume true.)
Converse: "If I bring an umbrella, then it is raining." (Not necessarily true — I might bring one for sun.)
Inverse: "If it is not raining, then I will not bring an umbrella." (Not necessarily true.)