2-3: Conditional Statements

1. 2-3: Conditional Statements

2. Conditional Statements • Conditional – formed by joining to statements with the words if and then: If p, then q. • Hypothesis – phrase immediately following if • Conclusion – phrase immediately following then

3. Example #1 • Identify the hypothesis and conclusion of each statement • If you get a 100% on your test, then you get an A. • Hypothesis: you get a 100% on your test • Conclusion: you get an A • The volleyball team will play in the playoffs if they are one of the top 4 teams. • Hypothesis: the volleyball team is one of the top 4 teams • Conclusion: they will play in the playoffs

4. Example #2 • Identify the hypothesis and conclusion of the following statements. Then write each statement in if-then form. • A five-sided polygon is a pentagon. • Hypothesis: a polygon has five sides • Conclusion: it is a pentagon • If a polygon has five sides, then it is a pentagon

5. Continued • An angle formed by perpendicular lines is a right angle. • Hypothesis: an angle is formed by perpendicular lines • Conclusion: it is a right angle • If an angle is formed by perpendicular lines, then it is a right angle.

6. Example #3 • Determine the truth value of the following statement for each set of conditions. • If Yukon rests for 10 days, his ankle will heal. • Yukon rests for 10 days, and he still has a hurt ankle. • The hypothesis is true, but the conclusion is false. • Since the result is not what was expected, the conditional statement is false

7. Continued • If Yukon rests for 10 days, his ankle will heal. • Yukon rests for 3 days, and he still has a hurt ankle. • The hypothesis is false, and the conclusion is false. The statement does not say what happens if Yukon only rests for 3 days. His ankle could possibly still heal. • We cannot say the statement is false. Therefore, the statement is true.

8. If Yukon rests for 10 days, his ankle will heal. • Yukon rests for 10 days, and he does not have a hurt ankle anymore. • The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he doesn’t have a hurt ankle. • Since what was stated is true, the conditional statement is true. • Yukon rests for 7 days, and he does not have a hurt ankle anymore. • The hypothesis is false, and the conclusion is true. The statement does not say what happens if Yukon only rests for 7 days. • We cannot say that the statement is false. Therefore, the statement is true. • RULES OF LOGIC: • A conditional is always false when the hypothesis is true and the conclusion is false.

9. Conditionals, Converses, etc. • Conditional – If p, then q. (given hypothesis and conclusion) • Converse – If q, then p. (switching the order of the hypothesis and conclusion) • Inverse – If not p, then not q. (negating both the hypothesis and conclusion) • Contrapositive – If not q, then not p. (negating both the hypothesis and conclusion of the converse) • RULES OF LOGIC: • The truth value of a conditional and its contrapositive are always the same. • The truth value of a converse and an inverse are always the same.

10. Example #4 • Write the converse, inverse, and contrapositive of the statement “All squares are rectangles.” Determine whether each statement is true or false. If the statement is false, give a counterexample. • First, write the conditional in if-then form. • If a shape is a square, then it is a rectangle • True • Converse: • If a shape is a rectangle, then it is a square. • False • Counterexample: a rectangle with a length of 2 and width of 4 is not a square

11. Inverse: • If a shape is not a square, then it is not a rectangle. • False • Counterexample: a 4-sided polygon with lengths 3, 3, 6, and 6 is not a square and is a rectangle. • Contrapositive: • If a shape is not a rectangle, then it is not a square. • True

12. Homework • Practice & Word Problem Practice