Conditional Statements
Overview
A conditional statement is a logical statement with two parts: a hypothesis and a conclusion, written in if-then form. This lesson is about taking one conditional statement and building its relatives — the converse, the inverse, and the contrapositive — by exchanging or negating the two parts, and about when those relatives must be true. It ends with biconditional statements, which combine a conditional and its converse and are exactly how definitions are written.
Vocabulary
- Conditional Statement
- A logical statement with two parts, a hypothesis and a conclusion, written in the form "if hypothesis, then conclusion." It is symbolized p → q, read "if p, then q."
- Hypothesis
- The "if" part of a conditional statement — the condition that is assumed. It is symbolized by p.
- Conclusion
- The "then" part of a conditional statement — the result that follows when the hypothesis is true. It is symbolized by q.
- Converse
- The statement formed by exchanging the hypothesis and the conclusion of a conditional statement: if q, then p (q → p). The converse of a true statement is not necessarily true.
- Inverse
- The statement formed by negating both the hypothesis and the conclusion of a conditional statement, keeping them in the same order: if not p, then not q (~p → ~q).
- Contrapositive
- The statement formed by exchanging the hypothesis and the conclusion of a conditional statement and negating both: if not q, then not p (~q → ~p). The contrapositive is always true when the conditional statement is true, and always false when it is false.
- Biconditional Statement
- A statement that combines a conditional statement and its converse with the phrase "if and only if" (p ↔ q). A biconditional is true only when both the conditional and its converse are true; definitions are biconditional statements.
Main Ideas
- Every conditional statement has a hypothesis (the "if" part, p) and a conclusion (the "then" part, q). Identifying the two parts correctly is the first step in every problem in this lesson.
- The converse exchanges p and q. A true conditional can have a false converse, so the converse always has to be checked on its own.
- The inverse negates p and q without exchanging them. Like the converse, the inverse is not necessarily true when the conditional is true.
- The contrapositive exchanges p and q and negates both. It is logically equivalent to the original conditional: one is true exactly when the other is.
- A biconditional statement (p ↔ q) packages a conditional and its converse together, so it can be used only when both directions are true — which is why every definition can be written as a biconditional.
Lesson Notes
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