Overview
This lecture explains how to construct the converse, inverse, and contrapositive of a conditional statement using examples and highlights their definitions and truth values.
Conditional Statements
- A conditional statement is written as "If p, then q" (if hypothesis, then conclusion).
- The hypothesis (p) is the part after "if"; the conclusion (q) is the part after "then".
- Symbol for negation ("not") indicates the opposite of a statement.
Converse, Inverse, and Contrapositive
- The converse switches hypothesis and conclusion: "If q, then p".
- The inverse negates both parts: "If not p, then not q".
- The contrapositive both reverses and negates: "If not q, then not p".
Truth Values and Biconditional Statements
- The converse and inverse always share the same truth value.
- The conditional and contrapositive always share the same truth value.
- A biconditional statement ("if and only if") is true if both the conditional and its converse are true.
Examples
- Original: "If you live in Los Angeles (p), then you live in California (q)".
- Converse: "If you live in California, then you live in Los Angeles" (false).
- Inverse: "If you don't live in Los Angeles, then you don't live in California" (false).
- Contrapositive: "If you don't live in California, then you don't live in Los Angeles" (true).
- Second Example: "If I am hungry, then I will eat pizza."
- Converse: "If I eat pizza, then I am hungry."
- Inverse: "If I am not hungry, then I will not eat pizza."
- Contrapositive: "If I do not eat pizza, then I am not hungry."
Key Terms & Definitions
- Conditional statement — If p, then q; links a hypothesis and conclusion.
- Hypothesis (p) — The “if” part of the conditional.
- Conclusion (q) — The “then” part of the conditional.
- Negation — The opposite of a statement ("not").
- Converse — Reverses the hypothesis and conclusion (If q, then p).
- Inverse — Negates both the hypothesis and conclusion (If not p, then not q).
- Contrapositive — Both reverses and negates (If not q, then not p).
- Biconditional statement — True when both conditional and converse are true.
Action Items / Next Steps
- Practice writing the converse, inverse, and contrapositive of sample conditional statements.
- Review truth values of each form and how they are logically related.