Conditional and Biconditional Statements

Conditional Statement: a statement written in if-then form. The “if” part is the hypothesis (p) and the “then” part is the conclusion (q). Symbol: 𝑝 → 𝑞 Read: “If p, then q.”

« Conditional statements can be True or False Counterexample: at least one fact or argument that indicates a statement or theorem is NOT true. Rewrite the conditional statement in if-then form. Identify the hypothesis and conclusion. Two points are collinear if they lie on the same line. If two points lie on the same line, then they are collinear. Hypothesis: Two points lie on the same line Conclusion: They are collinear Write a counterexample to show that the following conditional statement is false. 𝐼𝑓 𝑥! = 16, 𝑡ℎ𝑒𝑛 𝑥 = 4 Let 𝑥 = −4 The hypothesis is true because −4 ! = 16 The conclusion is false, so the conditional statement is false Converse: a conditional statement formed by switching the hypothesis and conclusion. Symbol: 𝑞 → 𝑝 Read: “If q, then p.” Statement: If you see lightning, then you hear thunder. Converse: If you hear thunder, then you see lightning.

Negation: writing the negative of a statement Statement: 𝑚 < 𝐴 = 30° Negation: 𝑚 < 𝐴 ≠ 30° < 𝐴 𝑖𝑠 𝑎𝑐𝑢𝑡𝑒 < 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑐𝑢𝑡𝑒 Inverse: when you negate the hypothesis and the conclusion of a conditional statement Contrapositive: when you negate the hypothesis and the conclusion of the converse of a conditional statement Original

If 𝑚 < 𝐴 = 30°, 𝑡ℎ𝑒𝑛 < 𝐴 𝑖𝑠 𝑎𝑐𝑢𝑡𝑒.

Inverse

If 𝑚 < 𝐴 ≠ 30°, 𝑡ℎ𝑒𝑛 < 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑐𝑢𝑡𝑒.

Converse

𝐼𝑓 < 𝐴 𝑖𝑠 𝑎𝑐𝑢𝑡𝑒, 𝑡ℎ𝑒𝑛 𝑚 < 𝐴 𝑖𝑠 30°.

Contrapositive

𝐼𝑓 < 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑐𝑢𝑡𝑒, 𝑡ℎ𝑒𝑛 𝑚 < 𝐴 𝑖𝑠 𝑛𝑜𝑡 30°.

Biconditional Statement: a statement written in if and only if form, where both the conditional and its converse are both true. Symbol: 𝑝 ↔ 𝑞 Read: “p if and only if q” Consider the following statement: 𝒙 = 𝟑 𝒊𝒇 𝒂𝒏𝒅 𝒐𝒏𝒍𝒚 𝒊𝒇 𝒙𝟐 = 𝟗 Is this a biconditional statement? Yes because it contains “if and only if” Is the statement true? Conditional Statement: If 𝑥 = 3, 𝑡ℎ𝑒𝑛 𝑥! = 9 Converse: 𝐼𝑓 𝑥! = 9, 𝑡ℎ𝑒𝑛 𝑥 = 3 The conditional is true, but the converse is not…..so the biconditional statement is false. Counterexample to the converse: −3 ! = 9