Inductive Reasoning. Inductive Reasoning. Inductive Reasoning. Inductive Reasoning. Logic (with Truth Tables) If-Then Statements

Transcription

1 Intro to Proofs (t-charts and paragraph) Table of Contents When asked a question you don't know the answer to: 1) You can take a known to be true. Using conjecture is Contents Bob is taller than Carl and Carl is taller than Phil. Make a conjecture of what comes next

2 Make a conjecture for number comes next: Make a conjecture for each of the following add 1, then 2, M is the midpoint of 1 Which conjecture can be drawn from the given 2 Which conjecture can be drawn from the given 3 Which conjecture can be drawn from the given 4 Which conjecture can be drawn from the given

3 5 Which conjecture can be drawn from the given In EFG, E is a right angle 6 Which conjecture can be drawn from the given is a right triangle. M is the midpoint of hypotenuse m m m m. This is why use theorems and 7 Using the given information, determine if the Given: A and B are on a number line. A is at 6 and B is 3 away. Conjecture: B is at 9 Conjecture: When a number is squared it is positive. 8 Using the given information, determine if the 9 Using the given information, determine if the Given: KM = MN Conjecture: M is the midpoint Given: A circle is drawn with center A and a point on the circle C Conjecture: AC is a radius of the circle

4 Using the given information, determine if the ` (with Truth Tables) Contents The Validity of a Statement Check to see if the statement is always true Trenton is the capital of NJ.

5 Negate the following statements, state the validity of the statement The symbol for negation is ~ The negation of statement p is ~p, said "not p" Compound Statements using And & OR And forms a conjunction conjunctions and disjunctions. Multiples of 2 Multiples of 3 Or forms a disjunction Multiples of 2 and 3 lie in the intersection Multiples of 2 or 3 lie in the union of the sets Note: When listing out the union each number appears once. Create a Venn Diagram for: p q

6 p q p q p p q And's and Or's can also be used with statements. q true? The number of rows needed is 2 where n is the number of variables used. In the example n=2, so 2 rows. For the second column alternate. false and q is true.

7 Examples: Make truth tables for the following. Examples: Make truth tables for the following. When making a truth table for ~p When making a truth table for ~p When is ~p When making a truth table for ~p p and q are both true p is true and q is false

8 When making a truth table for ~p When is ~p p, q, and r are all true p and q true, r false p and r true, q false p true, q and r false p false, q and r true p and r false, q true p and q true, r false Show that ~p q)= ~p ~(p true for all other cases. The statements are equivalent. Construct a truth table for the compound statement: Contents Oct 30-10:07 AM

9 , or Conditional Statements, are in sentences that given a certain condition the If today is Tuesday, then tomorrow is Wednesday. conclusion The hypothesis does not need to be first in the sentence, its the cause that leads to the effect. Underline the hypothesis with one line and the conclusion with 2. 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. You can go to the movies, if you clean your room. A conditional doesn't have to have if and then in it, just a Underline the hypothesis with one line and the conclusion with 2. A square has 4 right angles. We can check the validity of a conditional. The only way to show a conditional false is for T F We look for this pattern to have a testing to see if the statement is a lie. If you get a 100% on your next test, then I'll give you $1000. Find counterexamples for each of the following. If points A, B, and C lie in a plane, If a figure has 1 right angle, its a right triangle. If tomorrow we have no school, then today is Friday.

10 If today is May 32, then tomorrow is the 33rd. Given the conditional statement of p : If tomorrow is Tuesday, then today is Monday. : If today is Monday, then tomorrow is Tuesday. : If tomorrow is not Tuesday, then today is not Monday. : If today is not Monday, then tomorrow is not If all four are true, the statement is a definition. the same measure. : If angles have the same measure then they are : If angles are not vertical angles, then they do not If x = 2, then x If x 2 = 4, then x = 2. If x = 2, then x 2 = 4. if x 2 = 4, then x = 2. : If angles do not have the same measures

11 What is the inverse of: If a=0 then ab=0. If ab=0, then a=0. If ab=0, then a=0. Contents Make a conclusion using the Law of Detachment. q is true and p is true, then q is true. If the hypothesis of a true conditional statement is true, then the conclusion is also true. (To be able to conclude that q is true, both p must be true and the conditional must be true.) Example: Mary goes to the movies every Friday and Saturday night. Today is Friday. Using the Law of Detachment... "Today is Friday" satisfies the hypothesis of the conditional statement, so you can conclude that Mary will go to the movies tonight. 1) If it rains, then the game will be cancelled. If the 2nd statement is p, then q can be concluded. 1) If x=4 and y=5, then xy=20.

12 1) If an angle is a right angle, then its measure is 90 degrees. 1) If today is Saturday, then I'll wash my car. 2) If you win today, then you'll play tomorrow. Conclusion: I wash my car. 3) If a point is a midpoint, then it splits the segment into 2 congruent segments. 1) If angle A is obtuse, then it is greater than 90. 2) Angle A is not 90. 1) If an angle is bisected, then there are two angles with the same measure. Conclusion: Angle A is obtuse. 1) If a triangle has 3 equal sides,then it is equilateral. Law of Syllogism r is true, then p Conclusion: if these statements are true... then this statement is true...

13 Examples: Law of Syllogism: Write a new conditional statement given: If Rick takes chemistry this year, then Jesse will be Rick's lab partner. If Jesse is Rick's lab partner, then Rick will get an A in chemistry. If Rick takes chemistry this year, then Rick will get an A in chemistry. What about... Use the Law of Syllogism to write a new conditional statement for... If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all of its sides are congruent. Oct 30-2:57 PM Oct 30-3:00 PM One more... Use the Law of Syllogism to write a new conditional statement for... If x 2 > 25, then x 2 > 20 If x > 5, then x 2 > 25 Even though the conclusion of the second statement is the hypothesis of the first statement, you can still write a new conditional statement... If x > 5, then x 2 > 20 Make a conclusion using the Law of Syllogism. 1) If it rains, then the game will be cancelled. 2) If the game is cancelled, then we can go to the movies. If in the 2nd statement there is a repeat of the first statement use the same letter to label it. Look for the p r. 1) If a tree is an oak, then it grows very tall. 2) If a tree casts a large shadow, then it very tall. Oct 30-3:32 PM 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. 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. 2) If you win today, then you'll play tomorrow. If you play tomorrow, then you are in the championship game. 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

14 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. 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. 1) If you get a free soda at Burger Barn, then you won't have to go 2) If you spend $20 or more at Burger Barn, then you'll get a free Conclusion: If you spend $20 or more at Burger Barn,then you (t-charts and paragraph) Contents is true? The intersection of planes is line A line contains at least 2 points. Planes contain at least 3 non-collinear points. The intersection of 2 unique lines is a point. The intersection of 2 unique planes is a line. If 2 points from the same line are in a plane, the entire line is in A theorem is a statement proven true. The intersection of 2 unique planes is a line.

15 Line AB is the only line through both A and B Since AB lies in plane point C lies in plane The intersection of 2 unique planes is a line. The intersection of 2 unique planes is a line. Using a series of logical steps, like the Law of Detachment. Begin with what is given. Use definitions, theorems, and postulates to make new statements. When you've made the statement that you were asked to The intersection of 2 unique planes is a line. There are a few forms proofs can take. We are going to do T- Charts and Paragraph Proofs. T-Chart and justifications. Like writing an outline for a research and making them into sentences. 1) ABCD is a square 2) AB 3) AB=BC 1) Given 2) Definition of a Square 1) ABCD is a square 2) AB 3) AB=BC 1) Given 2) Definition of a Square Given ABCD is a square, we can use the definition of BC. By the definition of congruent, AB=BC

16 Given: AC Definition of Midpoint Definition of Congruen Now use the T-Chart to write a paragraph proof. BD, B is the midpoint of AC C is the midpoint of BD AB=CD Definition of Midpoint Definition of Congruen Substitution Make a reference sheet for yourself, if you haven't already started. What "justifications" have we learned so far? Def of Vertical Angles Def of Complementary Def of Perpendicular Def of Right Angle Points,Lines,and Planes Now use the T-Chart to write a paragraph proof. a(b + c) = ab + ac Reflexive Property Contents Algebraic Proofs Algebraic Proofs

17 Which property justifies the statement. Which property justifies the statement. If 5x = 5, Distributive Property Division Property of Equality Symmetric Property Distributive Property Algebraic Proofs Algebraic Proofs Which property justifies the statement. Which property justifies the statement. then 5x +35 = 12. Division Property of Equality Distributive Property Symmetric Property Division Property of Equality Distributive Property Algebraic Proofs Algebraic Proofs Which property justifies the statement. If 8 = x, solution a statement. Justify the step. Division Property of Equality Distributive Property Symmetric Property Division Property of Equality Distributive Property Symmetric Property Algebraic Proofs Algebraic Proofs

18 solution a statement. Justify the step. solution a statement. Justify the step. Division Property of Equality Distributive Property Symmetric Property Division Property of Equality Distributive Property Symmetric Property Algebraic Proofs Algebraic Proofs solution a statement. Justify the step. solution a statement. Justify the step. Division Property of Equality Distributive Property Symmetric Property Division Property of Equality Distributive Property Symmetric Property Algebraic Proofs Algebraic Proofs B is between A and C, then AB + BC = AC Cannot assume AB<BC, only that AB=4, BC=8, and AC=x AB=5, BC=x and AC=10 AB=x, BC=8 and AC= 12 AB=2x, BC=5x, and AC=28 Contents

19 There are 2 basic types of proofs with segments: Putting together smaller segments to make bigger segments AB=CD AC=BD 1) AB+BC=AC 2) BC+CD=BD 3) AB=CD 4) AB+BC=BC+CD 5) AC=BD 3) Given an equation, addition, substitution. Proofs of this type follow this basic pattern.

20 AC=MP for Segments DE=SE, EF=ET DF=ST F for Segments Def of Segment Bisector Def of Midpoint Def of Segment Bisector Def of Midpoint AC=BD AB=CD 3) Given 4) Substitution 5) Subtraction Property of Equality DF=ST X is the midpoint of DF X is the midpoint of ST DF=ST F for Segments Def of Segment Bisector Def of Midpoint an equation, substitute, and subtract. Proofs of this AC=XZ, AB=XY YZ=BC Z for Segments Def of Segment Bisector Def of Midpoint M

21 Cannot assume congruent angles, are congruent. congruent angles, are congruent. Perpendicular lines form right angles

22 F F F F F 60

23 There are 3 basic types of proofs with angles: Proving 2 angles are congruent because of the their relation to 1) Angle Addition Postulate 4) Addition Property of Equality the same thing (or congruent things) to both sides, substitution. Proofs of this type follow this basic pattern. M Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Angle Bisector Def of Vertical Angles Def of Complementary Complements of an Ang Supplements of an Angl Complements of an Ang Supplements of an Angl 1) Angle Addition Postulate Def of Angle Bisector Def of Vertical Angles Def of Complementary 2) Subtraction Prop 4)Subtraction Prop 5) Substitution Complements of an Ang Supplements of an Angl

24 Proving angles congruent because of their relation to a third 3) Substitution Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Angle Bisector Def of Vertical Angles Def of Complementary Complements of an Ang Supplements of an Angl Complements of an Ang Supplements of an Angl Jun 1-12:22 PM