Lesson 5.4 Conditional Statements pp Lesson 5.4 Conditional Statements pp

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. 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.

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 p  q. DefinitionDefinition

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. 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. 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.

pqpqpqpq T F T TT TF FT FF T Truth table for a conditional statement

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 = 5. 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 = 5. All three are true, since each hypothesis is false. The symbolic forms are F  F, F  F, and F  T. All three are true, since each hypothesis is false. The symbolic forms are F  F, F  F, and F  T.

A biconditional statement is a statement of the form “p, if and only if q,” (symbolized by p  q), which means p  q and q  p. DefinitionDefinition

Theorem 5.1 The conditional p  q is equivalent to the disjunction ~p or q. Theorem 5.1 The conditional p  q is equivalent to the disjunction ~p or q.

pqp  q  p  p  q(p  q)  (  p  q)T TTTFFTFFTTTFFTFFF TFTT FFFT TTTT TTTT

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.

The converse of a conditional statement is obtained by switching the hypothesis and conclusion. The converse of p  q is q  p. DefinitionDefinition

The inverse of a conditional statement is obtained by negating both the hypothesis and conclusion. The inverse of p  q is ~p  ~q. DefinitionDefinition

The contrapositive of a conditional statement is obtained by switching and negating the hypothesis and conclusion. The contrapositive of p  q is ~q  ~p. DefinitionDefinition

EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf we have a blizzard, then school will be canceled. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf we have a blizzard, then school will be canceled. ConverseIf school is canceled, then we had a blizzard.

EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf we have a blizzard, then school will be canceled. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf we have a blizzard, then school will be canceled. InverseIf we do not have a blizzard, then school will not be canceled.

EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf we have a blizzard, then school will be canceled. EXAMPLE 4 Write the converse, inverse, and contrapositive of the implication below. ImplicationIf 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. 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. 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. 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, p  q is equivalent to ~q  ~p. Theorem 5.2 Contrapositive Rule. A conditional statement is equivalent to its contrapositive. In other words, p  q is equivalent to ~q  ~p.

(p  q)  pqp  q~q~p~q  ~p(~q  ~p)T TF FTF 25. TFFTT FTFFT TFTTT TTTTT

Homework pp Homework pp

►A. Exercises Write the following statements in if-then form. 1.When I study my geometry, I get good grades. ►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. ►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. ►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. ►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. ►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. 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. ►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 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 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? 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? 33.icosahedron

■ Cumulative Review How many sides or faces does each figure have? 34.decagon ■ Cumulative Review How many sides or faces does each figure have? 34.decagon