Using Deductive Reasoning

1. Using Deductive Reasoning If – Then Statements; Converses

2. Vocabulary • If- then statements – conditional statements – conditionals • If it is sunny outside, then I will go out and play. • Hypothesis – it is sunny outside • Conclusion – I will go out and play • Geometry – If B is between A and C, then AB + BC = AC. • Generic statement – If p, then q.

3. Changing the conditional • Converse of a conditional is formed by interchanging the hypothesis and the conclusion. • If AB + BC = AC, then B is between A and C. • If q, then p. • Some true conditionals have false converses. • Converse of first statement. (If it is sunny outside, then I will go out and play.) • If ___________________, then __________. • Is this true?

4. Counterexample • An example that proves a statement to be false is called a counterexample. • If I go out to play does that mean it is sunny? • Ex. Identify the hypothesis, conclusion, and write the converse of the statement. Determine if it is true or false. If false give a counterexample. (5 parts) • If I dive in the water, then I will get wet. • If a + c = b + c, then a = b.

5. Other ways to write conditionals • The statements are not always written with the “if” clause first. If and only if (iff)– means that the conditional and the converse are true. They are called biconditional statements.

6. Practice • Tell whether the statement is true or false. Write the converse and determine if it is true or false. Can you provide a counterexample for each false statement? • If today is Friday, then tomorrow is Saturday. • If a number is divisible by 10, then it is divisible by 5. • If x < 0, then x² > 0.