IF-THEN STATEMENTS DAY 17

Transcription

1 IF-THEN STATEMENTS DAY 17 DEDUCTIVE REASONING: The process of using orderly statements to make logical conclusions. IF-THEN STATEMENTS: CONDITIONAL STATEMENTS CONDITIONAL CONVERSE (FLIPPED CONDITIONAL) IF P THEN Q IF Q THEN P P is the Hypothesis; Q is the Conclusion Q is the Hypothesis; P is the Conclusion If a figure is a triangle, then it s a polygon If a figure is a polygon, then it s a triangle TRUE FALSE INVERSE (CONDITIONAL with NOT) CONTRAPOSITIVE (FLIPPED CONDITIONAL WITH NO If NOT P, then NOT Q. If NOT Q, then NOT P. If a figure it s NOT a triangle, If a figure it s NOT a polygon, Then it s NOT a polygon then it s NOT a triangle FALSE TRUE CONDITIONAL and CONTRAPOSITIVE are LOGICALLY EQUIVALENT (either both True or both False CONVERSE and INVERSE are both LOGICALLY EQUIVALENT (either both True or both False) GOAL: ONLY USE GOOD DEFINITIONS TO PROVE SHORTCUTS.. A GOOD DEFINTION WHEN THE CONDITIONAL AND CONVERSE ARE BOTH TRUE. STATEMENT : Perpendicular lines form right angles. (A GOOD DEFINITION) CONDITIONAL CONVERSE IF two lines are perpendicular, IF two lines form right angles, THEN they form right angles. THEN the two lines are perpendicular TRUE TRUE IF AND ONLY IF STATEMENTS: When Conditional and its Converse are both TRUE BICONDITIONAL: Two lines are perpendicular IF AND ONLY IF they form right angles. STATEMENT: Two lines are perpendicular if they form right angles (A GOOD DEFINITION) STATEMENT: A triangle is a polygon (NOT A GOOD DEFINITON) CONDITIONAL IF a figure is a triangle, THEN it is a polygon. TRUE CONVERSE IF a figure is a polygon, THEN it is a triangle. FALSE IF AND ONLY IF: NOT POSSIBLE

2 PAIR WORK: Express as a Conditional and a Converse Statement. True or False. 1. All equilateral triangles are isosceles (NOT A GOOD DEFINITION) CONDITIONAL: If, THEN TRUE/FALSE CONVERSE: If, THEN TRUE/FALSE IF AND ONLY IF STATEMENT: NOT POSSIBLE 2. Obtuse triangles have an obtuse angle. (A GOOD DEFINITION) CONDITIONAL: If, THEN TRUE / FALSE CONVERSE: If, THEN TRUE / FALSE IF AND ONLY IF STATEMENT: A triangle is obtuse IF AND ONLY IF it has an obtuse angle.

3 C x is positive. 32. x)4 lf p, then q. Statement: The square of an integer is odd. 33. An integer is odd. p, then not q. If not Inverse: m are parallel. Lines intersect. Lines /DIAGRAMS ar.d mifdo 34. a polygon. it /isand then is MAKE a triangle, a not figure statement: USINGTrue EULER TO CONCLUSIONS polygon' not atriangle. then is it ais right LABC a figure is not a triangle, is a right Ifangle. 3s. inverse: FalseLA A polygon is regular. A polygon is equilateral. 36. DAY Alternate interior angles formed Related by lines I and m and transversal Summary / are congruent. of Lines / and m are parallel. If-Then Statements Given statement: If p, then q. Contrapositive: lf not q, then not P. 38. a. Given: dnll oc; ADll BC lf q, then p. Converse: Prove: 1A: /-C; LB: LD If not p, then not q. Inverse: b. Tell what is given and what is to be proved in the converse of converse.equivalent. thelogically contrapositive a proof ofare Thenitswrite A statement part (a). and its inverse. to an into and (b)or to its(a) converse have equivalent proved in parts logically not you Combine iswhat Ac.statement if-and-only-if statement. The relationships just summarized perto base conclusions on the contraposmit usdiagrams EULER itive of a true if-then statement bvt not on the converse or inverse. For example, suptrue: as an this statement accept we the pose if-then statement and its conbetween relationship To show diagrams to use circle verse, it is helpful All Olympic competitors,are athletes. (also called Venn diagrams or diagrams). (If Euler a person is an Olympic competitor, then statement p, we draw a circle named p. If p is To represent, person is an aathlete.) that true, we think of a point inside circle p. If p is false, we think of a 2-7 Converse, Contrapositive, Inverse t p is false. p is true. point outside circle p. In the diagram at the left below, a point that lies inside circlep must also lie inside circle q. In otherwords: If p,then q. Check to see that the middle diagram represents the converse: If q, then p. Check the diagram at the right 92 /also. Chapter 2 If 4, thenp. \f p, then q. p if and only if q. Compare the following if-then statements. Statement: lf p,lhen q. Parallel Lines and Planes Contrapositive: If not q, then not P. You already know that the diagram at the right represents "lf p, then q." The diagram also represents "If not Q, then not pi' because a point that isn't inside circle q can't be inside circlep either. Since the statement and its contrapositive are both true or else both false, they are called logically equivalent. The following statements are logically equivalent. True statement: If a figure is a triangle, then it is a polygon. Tiue contrapositive: If a hgure is not a polygon, then it is not a triangle. Since a statement and its contrapositive are logically equivalent, we may prove a statement by proving its contrapositive. Sometimes that is easier. There is one more conditional related to "If p, then q" that we will consider. A statement and its inuerse are not logically equivalent. Statement: lf p, then q. Inverse: If not p, then not q. True statement: False inverse: If If a figure is a triangle, then it is a polygon. a figure is not a triangle, then it is not a polygon' Summary of Related If-Then Statements / 9l

4 MORE EULER DIAGRAMS Ex. If competitors are Olympians then they are athletes This statement is paired with four different statements below. l. Giuen: lf p, then q. p Conclude: q All Olympic competitors are athletes. Ozzie is an Olympian. Ozzie is an athletes 2. Giuen: lf p, then q. rlot q Conclude: not p All Olympic competitors are athletes. Ned is not an athlete. Ned is not an Olympic athletes 3. Giuen: lf p, then q. q No conclusion follows. All Olympic competitors are athletes. Anne is an athlete. Anne might be an Olympic competitor or she might not be. 4. Giuen: lf p,lhen q. Irot p No conclusion follows. All Olympic competitors are athletes. Nancy is not an Olympic competitor. Nancy might be an athlete or she might not

5 CW#18 / HW#18 1. Given: All senators are at least 30 years old. a. Reword this statement in if-then form. Conditional: If someone is a senator then he/she is at least 30 yrs old Contrapositive: If someone is younger than 30 yrs old then he/she is not a senator b. Make a circle diagram to illustrate the Conditional statement. c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no. (Hint; Only make conclusions per the Conditional or the Contrapositive Statements) 1. Jose Avila is 48 years old. 2. Rebecca Castelloe is a senator 3. Constance Brown is not a senator. 4. Ling Chen is 29 years old. 2. Given: When it is not raining, I am happy a. Reword this statement in if-then form. b. Make a circle diagram to illustrate the statement. c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no. 1. I am not happy. 2. It is not raining. 3. I am overjoyed. 4. It is raining. 3. Given: All my students love geometry a. Reword this statement in if-then form. b. Make a circle diagram to illustrate the statement. c. If the given statement is true, what can you conclude from each of the following additional statements? If no conclusion is possible, say no. 1. Stu is my student. 2. Luis loves geometry. 3. Stells is not my student. 4. George does not love geometry.

6 INDIRECT REASONING /PROOFS DAY 19 INDIRECT REASONING: 1. Uses the idea that if a CONDITIONAL is TRUE, then its CONTRAPOSITIVE is also TRUE. CONDITIONAL: IF P THEN Q CONTRAPOSITIVE: IF NOT Q THEN NOT P 2. Uses the CONTRAPOSITIVE as the INDIRECT REASONING USING INDIRECT REASONING Explain how you would know if a driver applied the brakes. STATEMENT: CONDITIONAL: A car leaves skid marks when it applies the brakes. If a car leaves skid marks then it has applied the brakes CONTRAPOSITIVE: If a car does not apply the brakes, then it will not leave skid marks. INDIRECT REASONING: If a car does not apply the brakes, then it will not leave skid marks. Skid marks were left by the car. Therefore, the car must have applied the brakes. USING INDIRECT REASONING: Explain why ice is forming on the sidewalk in front of Toni s house. STATEMENT: CONDITIONAL: Ice forms when it is 32F or below. If ice forms then the temperature is 32F or below. CONTRAPOSITIVE: If the temperature is more than 32F, then ice will not form on the sidewalk. INDIRECT REASONING: If the temperature is more than 32F, then ice will not form on the sidewalk.. Ice is forming on the sidewalk. Therefore, the temperature must be 32F or less.

7 PAIR WORK: USING INDIRECT REASONING Johnnie is too lazy to create flash cards. Explain how you know he isn t going to get an A STATEMENT: CONDITIONAL: Every student who gets an A in Geometry creates and uses flash cards. If THEN CONTRAPOSITIVE: IF THEN INDIRECT REASONING: INDIRECT PROOFS: PROVING BY CONTRADICTION 1. Assume temporarily that the conclusion is not true. 2. Reason logically until you reach a contradiction of a known fact 3. Therefore, the temporary assumptions must be false and what needs to be proven must be true Given (Hypothesis): n is an integer and n 2 is even Prove (Conclusion): n is even Indirect Proof: 1. Assume temporarily that n is not even. 2. Then n is odd, and n X n = odd. This contradicts the given information that n 2 is even. 3. Therefore, that n is not even must be false.

8 CW#19 / HW#19 What is the first sentence of an indirect proof of the statement shown? 1. Triangle ABC is equilateral. 2. Doug is Canadian. 3. a b 4. Kim isn t a violinist. 5. Write an Indirect Proof Given (Hypothesis): A triangle Prove (Conclusion): There can be at most 1 right angle a. Assume temporarily that b. Then c. Therefore 6. Write an Indirect Proof Given (Hypothesis): Fresh skid marks appear behind a green car at the scene Prove (Conclude); The car must have applied the brakes. a. Assume temporarily that b. Then c. Therefore

9 7. Write an Indirect Proof Given (Hypothesis): Ice is forming on the side walk. Prove (Conclude); The temperature outside must be 32F or less. a. Assume temporarily that b. Then c. Therefore What conclusions, if any, can you make from each pair of statements? 8. There are three types of drawbridges; bascule, lift and swing. This drawbridge doesn t swing or lift. Conclusion: 9. If this were the day of the party, our friends would be home. No one is home. Conclusion: 10. Every traffic controller in the world speaks English on the job. Sumiko does not speak English Conclusion: 11. If non-vertical lines are perpendicular, then the product of their slops is -1. The product of the slopes of non-vertical lines is not -1. Conclusion: