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Lesson 5.4 Conditional Statements pp. 176-181 Objectives: 1. To define and write conditional statements. 2. To define biconditional statements. 3. To define and symbolize the inverse, converse, and contrapositive of a conditional. Definition A conditional statement is a statement of the form “If p, then q,” where p and q are statements. The notation for this conditional statement is pq. Notice that the p represents the “if” statement, called the hypothesis, and the q represents the “then” statement, called the conclusion. EXAMPLE 1 Write the following statements in if-then form. a. There are no clouds in the sky, so it is not raining. If there are no clouds in the sky, then it is not raining. EXAMPLE 1 Write the following statements in if-then form. b. School will be canceled if a blizzard hits. If a blizzard hits, then school will be canceled. Truth table for a conditional statement p q pq T T T T F F F T T F F T EXAMPLE 2 Which is true? If horses had wings, then horses could fly. If ocean water is grade-A milk, then ocean water is nourishing beverage. If whales walk, then 4 + 1 = 5. All three are true, since each hypothesis is false. The symbolic forms are F F, F F, and F T. Definition A biconditional statement is a statement of the form “p, if and only if q,” (symbolized by pq), which means pq and qp. Theorem 5.1 The conditional pq is equivalent to the disjunction ~p or q. p T T F F q pq p pq (pq)(pq) T T F T T F F F F T T T T T T F T T T T EXAMPLE 3 Change the following conditional statement to a disjunction. If a child disobeys, then he will be disciplined. A child obeys, or the child will be disciplined. Definition The converse of a conditional statement is obtained by switching the hypothesis and conclusion. The converse of pq is qp. Definition The inverse of a conditional statement is obtained by negating both the hypothesis and conclusion. The inverse of pq is ~p~q. Definition The contrapositive of a conditional statement is obtained by switching and negating the hypothesis and conclusion. The contrapositive of pq is ~q~p. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. Implication If we have a blizzard, then school will be canceled. Converse If school is canceled, then we had a blizzard. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. Implication If we have a blizzard, then school will be canceled. Inverse If we do not have a blizzard, then school will not be canceled. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. Implication If we have a blizzard, then school will be canceled. Contrapositive If school is not canceled, then we did not have a blizzard. Practice: Give the converse, inverse, and contrapositive of “if two angles are congruent, then they have the same measure.” Converse: If two angles have the same measure, then they are congruent. Practice: Give the converse, inverse, and contrapositive of “if two angles are congruent, then they have the same measure.” Inverse: If two angles are not congruent, then they do not have the same measure. Practice: Give the converse, inverse, and contrapositive of “if two angles are congruent, then they have the same measure.” Contrapositive: If two angles do not have the same measure, then they are not congruent. Theorem 5.2 Contrapositive Rule. A conditional statement is equivalent to its contrapositive. In other words, pq is equivalent to ~q~p. 25. (pq) p q pq ~q ~p ~q~p (~q~p) TT T F F T T TF F T F F T FT T F T T T FF T T T T T Homework pp. 180-181 ►A. Exercises Write the following statements in if-then form. 1. When I study my geometry, I get good grades. If I study my geometry, then I get good grades. ►A. Exercises Write the following statements in if-then form. 5. The flowers bloom because the sun shines. If the sun shines, then the flowers will bloom. ►A. Exercises State whether the conditionals are true or false. 9. If our pig is a clean animal, then we will keep it in the house. True ►A. Exercises State whether the conditionals are true or false. 11. The roads will be slick if and only if there is ice on the roads. False ►B. Exercises Write each biconditional as two conditionals. 17. You will get an A in geometry if and only if you study hard. If you study hard, then you will get an A in geometry. If you get an A in geometry, then you studied hard. ►B. Exercises Change each conditional to a disjunction. 21. If the stoplight is green, then you can go. The stoplight is not green, or you can go. ■ Cumulative Review True/False 30. If a triangle is isosceles, then it is equilateral. ■ Cumulative Review True/False 31. Theorem 1.3 could be worded “if two lines intersect, then there is a plane containing them; and if two lines intersect, then there is at most one plane containing them.” ■ Cumulative Review How many sides or faces does each figure have? 32. heptagon ■ Cumulative Review How many sides or faces does each figure have? 33. icosahedron ■ Cumulative Review How many sides or faces does each figure have? 34. decagon