The document discusses conditional statements and their components. It provides examples of identifying the hypothesis and conclusion of conditional statements. It also discusses determining the truth value of conditional statements based on whether the hypothesis and conclusion are both true, false, or one is true and one is false. Examples are given for writing the converse, inverse, and contrapositive of conditional statements and determining if they are true or false, providing counterexamples when false.

1. Conditional Statements Conditional - Conditionals are formed by joining two statements with the words if and then: If p, then q. The if-statement is the hypothesis and the then-statement is the conclusion . Conditional statements are either true conditionals or false conditionals. A conditional is a false conditional when the hypothesis is true and the conclusion is false. A conditional can be shown to be false by using a counterexample.

3. Identify Hypothesis and Conclusion A. Identify the hypothesis and conclusion of the following statement. If a polygon has 6 sides, then it is a hexagon.

4. Identify Hypothesis and Conclusion A. Identify the hypothesis and conclusion of the following statement. Answer: Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon If a polygon has 6 sides, then it is a hexagon. If a polygon has 6 sides, then it is a hexagon. hypothesis conclusion

5. Identify Hypothesis and Conclusion B. Identify the hypothesis and conclusion of the following statement. Tamika will advance to the next level of play if she completes the maze in her computer game.

6. Identify Hypothesis and Conclusion B. Identify the hypothesis and conclusion of the following statement. Tamika will advance to the next level of play if she completes the maze in her computer game. Answer: Hypothesis: Tamika completes the maze in her computer game Conclusion: she will advance to the next level of play

7. Write a Conditional in If-Then Form B. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. A five-sided polygon is a pentagon.

8. Write a Conditional in If-Then Form B. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. A five-sided polygon is a pentagon. Answer: Hypothesis: a polygon has five sides Conclusion: it is a pentagon If a polygon has five sides, then it is a pentagon.

9. A. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Truth Values of Conditionals Yukon rests for 10 days, and he still has a hurt ankle.

10. A. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Answer: Since the result is not what was expected, the conditional statement is false. Truth Values of Conditionals The hypothesis is true, but the conclusion is false. Yukon rests for 10 days, and he still has a hurt ankle.

11. B. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Truth Values of Conditionals Yukon rests for 3 days, and he still has a hurt ankle.

12. B. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true. Truth Values of Conditionals 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. Yukon rests for 3 days, and he still has a hurt ankle.

13. C. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Truth Values of Conditionals Yukon rests for 10 days, and he does not have a hurt ankle anymore.

14. C. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Answer: Since what was stated is true, the conditional statement is true. Truth Values of Conditionals The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he does not have a hurt ankle. Yukon rests for 10 days, and he does not have a hurt ankle anymore.

15. D. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Truth Values of Conditionals Yukon rests for 7 days, and he does not have a hurt ankle anymore.

16. D. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true. Truth Values of Conditionals The hypothesis is false, and the conclusion is true. The statement does not say what happens if Yukon only rests for 7 days. Yukon rests for 7 days, and he does not have a hurt ankle anymore.

17. Conditionals, Converses, etc Conditional - If p, then q. Converse - If q, then p. Inverse – If not p, then not q. Contrapositive – If not q, then not p. Counterexample – If p, then not q. Rules of Logic: The truth value of a converse may or may not be the same as that of its conditional. The truth value of a conditional and its contrapositive are always the same. Likewise for a converse and an inverse.

19. Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Related Conditionals

20. Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Related Conditionals Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. First, write the conditional in if-then form. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with and w = 4 is not a square.

21. Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square. The contrapositive is formed by negating the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true. Related Conditionals