Slide 54 / 156 Equivalent statements Truth Tables can also be used to show 2 statements are equivalent. Example: Show that ~(p # q)= ~p ∪ ~q ~ ~p ∪ ~ p q ~p ~q p q p # q (p # q) q T T F F F T T T F T F F T T T F F T F T T F T F T F T F F T T T F F F T Both statements false only if both p and q are true. They are true for all other cases. The statements are equivalent.
Slide 55 / 156 Show that ~p # ~q is equivalent to ~(p ∪ q).
Slide 56 / 156 If-Then Statements Return to Table of Contents
Slide 57 / 156 If-Then Statements, or Conditional Statements, are in sentences that given a certain condition the following event happens. The if part is called hypothesis or antecedent ( If today isTuesday... ) The then part is called conclusion or consequent ( ... then tomorrow is Wednesday ) Note: The hypothesis does not need to be first in the sentence, its the cause that leads to the effect. ( You can go to the movies, if you clean your room.)
Slide 58 / 156 Underline the hypothesis with one line and the conclusion with two lines. If add 2 and 2, then you'll get 4. If it rains tomorrow, then the picnic will be cancelled. You can use the car, if you'll put gas in it. If the Jets win or the Bengals lose, then the Jets make the playoffs. I'll buy dinner next time, if you buy dinner this time.
Slide 59 / 156 A conditional doesn't have to have if and then in it, just a hypothesis and a conclusion. Underline the hypothesis with one line and the conclusion with two lines. Rewrite the sentence in "if then" form. A square has 4 right angles. Bobby is in big trouble. On a windy day you can fly a kite.
Slide 60 / 156 The symbol notation for conditionals is p ⇒ q,which is read "p implies q" We can check the validity of a conditional. The only way to show a conditional false is for T ⇒ F p q p ⇒ q T T T We look for this pattern to have T F F a counter example. F T T These cases seem illogical, but F F T we're testing to see if the statement is a lie. If you get a 100% on your next test, then I'll give you $1000. The only pay to prove me a liar is to get a 100% and for me not to pay up.
Slide 61 / 156 Find counterexamples for each of the following. (1) If it's summer, then it must be July.
Slide 62 / 156 Find counterexamples for each of the following. (2) If you get a 90% on the test, then you'll get an A for the year.
Slide 63 / 156 Find counterexamples for each of the following. (3) If a figure has 1 right angle, its a right triangle.
Slide 64 / 156 Find counterexamples for each of the following. (4) If today is the 30th, then tomorrow is the 31st.
Slide 65 / 156 Find counterexamples for each of the following. (5) If tomorrow we have no school, then today is Friday.
Slide 66 / 156 25 Is this conditional T or F. If false, be ready with a counterexample. If points A, B, and C lie in a plane, then they are collinear. True False
Slide 67 / 156 26 Is this conditional T or F. If false, be ready with a counterexample. If points A, B, and C lie on a line, then B is between A and C. True False
Slide 68 / 156 27 Is this conditional T or F. If false, be ready with a counterexample. If today is May 32, then tomorrow is the 33rd. True False
Slide 69 / 156 Converse, Inverse, and Contrapositive Given the conditional statement of p ⇒ q there are 3 associated conditionals Converse: q ⇒ p Inverse: ~p ⇒ ~q Contrapositive: ~q ⇒ ~p The validity of a conditional and its contrapositive are equivalent. The validity of the converse and inverse are the same. If all four are true, the statement is a definition.
Slide 70 / 156 Examples: Conditional: If tomorrow is Tuesday, then today is Monday. Converse: If today is Monday, then tomorrow is Tuesday. Inverse: If tomorrow is not Tuesday, then today is not Monday. Contrapositive: If tomorrow is not Monday, then today is not Tuesday. Which statements are true?
Slide 71 / 156 Examples: Conditional: If angles are vertical angles, then they have the same measure. Converse: If angles have the same measure then they are vertical angles. Inverse: If angles are not vertical angles, then they do not have the same measure. Contrapositive: If angles do not have the same measures then they are not vertical angles. Which statements are true?
Slide 72 / 156 Example: Conditional: If x = 2, then x 2 = 4. Converse: Inverse: Contrapositive: Which of these statements are true?
Slide 73 / 156 28 What is the inverse of: If a=0 then ab=0. If a≠0,then ab≠0. A If ab≠0, then a≠0. B If ab=0, then a=0. C If a≠0, then ab=0. D
Slide 74 / 156 29 What is the converse of: If a=0 then ab=0. If a≠0,then ab≠0. A If ab≠0, then a≠0. B If ab=0, then a=0. C If a≠0, then ab=0. D
Slide 75 / 156 30 What is the validity of the contrapositive of: If a=0, then ab=0. True False
Slide 76 / 156 Deductive Reasoning Return to Table of Contents
Slide 77 / 156 Law of Detachment Deductive Reasoning can be used make valid conclusions involving conditionals. If p ⇒ q is true and p is true, then q is true. (To be able to conclude that q is true, both p must be true and the conditional must be true.)
Slide 78 / 156 Make a conclusion using the Law of Detachment. Statements 1) If it rains, then the game will be cancelled. 2) Its raining. Try underlining parts of the conditional and label them p and q. If the 2nd statement is p, then q can be concluded. Statements 1) If x=4 and y=5, then xy=20. 2) xy=20.
Slide 79 / 156 1) If an angle is a right angle, then its measure is 90 degrees. Angle ABC is a right angle. Conclusion_________________ 2) If you win today, then you'll play tomorrow. You won today. Conclusion_______________________ 3) If a point is a midpoint, then it splits the segment into 2 congruent segments. AM=MB. Conclusion____________________________
Slide 80 / 156 31 Does the following conclusion follow from the first two statements? 1) If today is Saturday, then I'll wash my car. 2) Today is Saturday. Conclusion: I wash my car. True False
Slide 81 / 156 32 Does the following conclusion follow from the first two statements? 1) If angle A is obtuse, then it is greater than 90. 2) Angle A is not 90. Conclusion: Angle A is obtuse. True False
Slide 82 / 156 33 Does the following conclusion follow from the first two statements? 1) If an angle is bisected, then there are two angles with the same measure. 2) Angle A and angle B have the same measure. Conclusion: They were made with an angle bisector. True False
Slide 83 / 156 34 Does the following conclusion follow from the first two statements? 1) If a triangle has 3 equal sides,then it is equilateral. 2) # ABC has three equal sides. Conclusion: # ABC is equilateral. True False
Slide 84 / 156 Law of Syllogism Deductive Reasoning can be used make valid conclusions involving conditionals. If p ⇒ q is true and q ⇒ r is true, then p ⇒ r is true.
Slide 85 / 156 Make a conclusion using the Law of Syllogism. Statements 1) If it rains, then the game will be cancelled. 2) If the game is cancelled, then we can go to the movies. Try underlining parts of the conditional and label them p and q. If in the 2nd statement there is a repeat of the first statement use the same letter to label it. Look for the p ⇒ q and q ⇒ r so we can conclude p ⇒ r. Statements 1) If a tree is an oak, then it grows very tall. 2) If a tree casts a large shadow, then it grows very tall.
Slide 86 / 156 1) If an angle is a right angle, then its measure is 90 degrees. If an angle is 90 degrees, then it is not acute or obtuse. Conclusion_________________
Slide 87 / 156 2) If you win today, then you'll play tomorrow. If you play tomorrow, then you are in the championship game. Conclusion_______________________
Slide 88 / 156 3) If a point is a midpoint, then it splits the segment into 2 congruent segments. If a point is a midpoint then it is collinear with the endpoints of a segment. Conclusion____________________________
Slide 89 / 156 35 Does the following conclusion follow from the first two statements? 1) If a triangle has 3 equal sides, then it is equilateral. 2) If a triangle is equilateral, then it also has 3 equal angles. Conclusion: If a triangle has 3 equal sides, then it has 3 equal angles. True False
Slide 90 / 156 36 Does the following conclusion follow from the first two statements? 1) If Carol goes to Rutgers, then she will major in accounting. 2) If Carol majors in accounting, then she will get a good job. Conclusion: If Carol has a good job,then she went to Rutgers. True False
Slide 91 / 156 37 Does the following conclusion follow from the first two statements? 1) If figure is a circle, then it measures 360 degrees. 2) If a figure is a square, then its angles add to 360 degrees Conclusion: If a figure is a square, then it is a circle. True False
Slide 92 / 156 38 Does the following conclusion follow from the first two statements? 1) If you get a free soda at Burger Barn, then you won't have to go to the convenience store. 2) If you spend $20 or more at Burger Barn, then you'll get a free soda. Conclusion: If you spend $20 or more at Burger Barn,then you won't have to go to the convenience store. True False
Slide 93 / 156 Intro to Proofs (t-charts and paragraph) Return to Table of Contents
Slide 94 / 156 A postulate is a statement that is accepted as true. A line Through any Through any 3 contains at 2 points there points not least 2 points. is exactly one collinear, there line. is exactly one plane. The intersection Planes contain of 2 unique at least 3 non- lines is a collinear point. points. A theorem is a statement proven true.
Slide 95 / 156 39 Which of the following postulates justifies the statement is true? The intersection of planes P and R is line l. A Through any 2 points there is exactly 1 line. B A line contains at least 2 points. P l R C A plane contains at least 3 noncollinear points. D The intersection of 2 unique lines is a point. E The intersection of 2 unique planes is a line.
Slide 96 / 156 40 Which of the following postulates justifies the statement is true? Line AB is the only line through both A and B A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C A line contains at least 2 points. A D The intersection of 2 unique lines is a point. E The intersection of 2 unique planes is a line. B
Slide 97 / 156 41 Which of the following postulates justifies the statement is true? Since AB lies in plane S , point C lies in plane S. A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C S C A line contains at least 2 points. B A D A plane contains at least 3 noncollinear points. E The intersection of 2 unique planes is a line. F If 2 points from the same line lie in a plane, the entire line is in the plane.
Slide 98 / 156 42 Which of the following postulates justifies the statement is true? The bottom of a three-legged stool will sit flat on the floor. A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C A line contains at least 2 points. D A plane contains at least 3 noncollinear points. E The intersection of 2 unique lines is a point. F The intersection of 2 unique planes is a line.
Slide 99 / 156 Questions about Proofs So how does a postulate become a theorem? Using a series of logical steps, like the Law of Detachment. How do you begin? Begin with what is given. Use definitions, theorems, and postulates to make new statements. So just make statements? Each statements needs a justification, like a lawyer presenting a case needs evidence.
Slide 100 / 156 Questions about Proofs When do I stop making new statements? When you've made the statement that you were asked to prove. I've heard there is a special form it has to follow? There are a few forms proofs can take. We are going to do T- Charts and Paragraph Proofs.
Recommend
More recommend
