Overview
This lecture introduces conditional statements in logic, explains related conditional forms (converse, inverse, contrapositive), explores negations, and presents biconditional statements, all foundational to reasoning and proofs in geometry.
Conditional Statements
- A conditional statement is a logical statement with two parts: the hypothesis (p) and the conclusion (q).
- Written in "if-then" form: if p, then q (symbolically: p → q).
- Example: If someone is a member of the soccer team, then they have practice today.
Negation
- Negation means stating the opposite of a statement.
- To negate, insert or remove "not": the negation of "the ball is blue" is "the ball is not blue."
- Do not replace terms with opposites (e.g., blue with red); use "not" appropriately.
Related Conditionals
- Converse: Switch hypothesis and conclusion. If q, then p.
- Inverse: Negate both hypothesis and conclusion. If not p, then not q.
- Contrapositive: Switch and negate both parts. If not q, then not p.
- Example using "You are a guitar player" (p) and "You are a musician" (q):
- Conditional: If you are a guitar player, then you are a musician. (True)
- Converse: If you are a musician, then you are a guitar player. (False)
- Inverse: If you are not a guitar player, then you are not a musician. (False)
- Contrapositive: If you are not a musician, then you are not a guitar player. (True)
Biconditional Statements
- A biconditional statement uses "if and only if" and is valid when both the conditional and converse are true.
- Format: p if and only if q.
- Example: Two lines intersect to form a right angle if and only if they are perpendicular lines.
Key Terms & Definitions
- Conditional Statement — A logical statement in the form "if p, then q."
- Hypothesis (p) — The "if" part of a conditional statement.
- Conclusion (q) — The "then" part of a conditional statement.
- Negation — The opposite of a statement, typically by adding or removing "not."
- Converse — Swaps hypothesis and conclusion: if q, then p.
- Inverse — Negates both parts: if not p, then not q.
- Contrapositive — Swaps and negates both parts: if not q, then not p.
- Biconditional Statement — A statement combining a true conditional and its converse with "if and only if."
Action Items / Next Steps
- Practice writing conditional, converse, inverse, and contrapositive forms for provided statements.
- Identify if statements are true or false.
- Prepare for further topics in reasoning and proofs by reviewing these logical forms.