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Conditional Statements

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 .

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Conditional Statements

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  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.

  2. 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.

  3. hypothesis conclusion 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. If a polygon has 6 sides, then it is a hexagon. Answer: Hypothesis: a polygon has 6 sidesConclusion: it is a hexagon

  4. 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.

  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. Answer: Hypothesis: Tamika completes the maze in her computer gameConclusion: she will advance to the next level of play

  6. 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.

  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. Answer: Hypothesis: a polygon has five sidesConclusion: it is a pentagonIf a polygon has five sides, then it is a pentagon.

  8. Truth Values of Conditionals A. 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.

  9. Truth Values of Conditionals A. 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. Answer: Since the result is not what was expected, the conditional statement is false.

  10. Truth Values of Conditionals B. 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 3 days, and he still has a hurt ankle.

  11. Truth Values of Conditionals B. 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 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. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.

  12. Truth Values of Conditionals C. 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 does not have a hurt ankle anymore.

  13. Truth Values of Conditionals C. 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 does not have a hurt ankle anymore. The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he does not have a hurt ankle. Answer: Since what was stated is true, the conditional statement is true.

  14. Truth Values of Conditionals D. 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 7 days, and he does not have a hurt ankle anymore.

  15. Truth Values of Conditionals D. 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 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. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.

  16. 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.

  17. Related Conditionals 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.

  18. 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. Related Conditionals 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. First, write the conditional in if-then form. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. Write the converse by switching the hypothesis and conclusion of the conditional.

  19. Related Conditionals 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.

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