Chapter Two: Reasoning and Proof

1. Chapter Two:Reasoning and Proof Section 2-1: Conditional Statements

2. Objectives • To recognize conditional statements. • To write converses of conditional statements.

3. Vocabulary • Conditional • Hypothesis • Conclusion • Truth Value • Converse

4. Conditional • A conditional is an “if……then……” statement. • Every conditional has two parts: • A hypothesis • A conclusion

5. Hypothesis • The part of the conditional that follows “if” in an “if….then….” statement is the hypothesis.

6. Conclusion • The part of the conditional that follows “then” in an “if….then….” statement is the conclusion.

7. Identifying the hypothesis and the conclusion • If x - 38 = 3, then x = 41 • Hypothesis:____________ • Conclusion:____________ • If Beca beats Emmaus, then they will win the LVC softball championship. • Hypothesis:____________ • Conclusion:____________

8. Writing a Conditional • The following statement is not in conditional form: • An integer that ends in 0 is divisible by 5. • We can re-write it in conditional form: • If an integer ends in 0, then it is divisible by 5.

9. Re-write the following statements in conditional form • A triangle has three sides. • An honor roll student must pass conduct.

10. Truth Value • A conditional can have a truth value of either true or false. • To show that a conditional is true, you must show that for every time the hypothesis is true, the conclusion is also true. • To show that a conditional is false, you must show only one case where the hypothesis is true and the conclusion is false. • To show a conditional is false we need to provide a counterexample.

11. Finding a Counterexample • If it is February, then there are 28 days in the month. • If a number is prime, then it is odd.

12. Using Venn Diagrams to Write Conditionals Residents of Pennsylvania • The Venn Diagram illustrates the conditional statement: • If you live in Bethlehem, then you live in Pennsylvania. Residents of Bethlehem

13. Converse • The converse of a conditional switches the hypothesis and the conclusion. • In can be possible for a conditional and its converse to have different truth values.

14. Writing the Converse of a Conditional • If x =12 then 2x = 24 • True statement • Converse: If 2x=24 then x = 12– in this case the converse is also true • If lines are parallel, then they do not intersect. • True statement • Converse: If two lines do not intersect, then they are parallel– this statement is false, the lines could be skew lines.

15. Symbolic Form • Conditional statement: p q is read, “if p then q” • p is the hypothesis • q is the conclusion • Converse: q p is read, “if q then p”