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2.2 Logic and Venn Diagrams
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true Statement: is a sentence that is either ____________ or ____________ called a _________ _________. Negation: has the ___________ meaning and the _________ __________ ________. Compound statement: two or more statements joined by the word _________ or _______. Conjunction: compound statement using the word ___________. false Truth value opposite opposite truth value and or and Statement Symbols Negation ~p, read not p Conjunction p q, read p and q Disjunction p q, read p or q
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Example 1 Use the following statement to write a compound statement for each conjunction. Then find the truth value. p: The figure is a triangle. q: The figure has two congruent sides. r: The figure has three acute angles. p q ________________________________________ not p and not r ______________________________________________________________ The figure is a triangle and the figure has two congruent sides. Truth Value: true The figure is not a triangle and the figure does not have three acute angles. Truth Value: true
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Disjunction: compound statement that uses the word _________.
Example 2 Use the following statement to write a compound statement for each disjunction. Then find the truth value. p: January is a fall month. q: January has only 30 days. r: January 1 is the first day of a new year. r or p b) q ~r c) p ~ q January 1 is the first day of a new year or January is a fall month. T January has only 30 days or January 1 is not the first day of a new year. F January is a fall month or January does not have only 30 days. T
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Truth value: a method for organizing the __________ __________ of negations and compound statements.
values Example 3 Fill in the truth table. a) b) Negation p ~p T F Conjunction p q p q T F c) Disjunction p q p q F T T T T T F T F T F F T T F F F F
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Example 4 Construct a truth table to the compound statement.
a) ~p q p q ~p ~p V q T T F T T F F F F T T T F F T T b) ~p q p q ~p ~q ~p ~q T T F F F T F F T F F T T F F F F T T T
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Example 5 The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. Tap Jazz 9 Ballet 29 How many students are enrolled in all three classes?______________ How many students are enrolled in tap or ballet?______________ How many students are enrolled in jazz and ballet, but not tap?___________ 9 121 25
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b. How many students play drums or synthesizer?
all three instruments? b. How many students play drums or synthesizer? c. How many students play both drums and guitar? 2 47 13
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7 students are on both the soccer and hockey teams
The athletic director needs to schedule a bus for a game/meet. Listed below are the number of members on teams. 20 students play hockey 40 students run track 46 students play soccer 7 students are on both the soccer and hockey teams 5 students are on both track and hockey teams 8 students are on the track and soccer teams 3 students are on all three teams How many students participate only in soccer? soccer track hockey 2 30 11 3 5 4 34 34
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32 students bike to school 21 students walk to school
The principal at an Elementary School needs to determine the modes of transportation for his students. 31 students come by car 32 students bike to school 21 students walk to school 5 students come by car and bike 8 students walk and bike to school 3 students come by car and walk 1 student comes by car and walks and bikes How many students only bike to school? 4 24 20 1 2 7 11 20
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2.3 Conditional Statements
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Conditional Statement: a statement that can be written in ____-____ _____. Hypothesis: part of the conditional statement following ___________. Conclusion: part of the conditional statement following ___________. If then form if then
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Example 1 Identify the hypothesis and conclusion of each conditional statement.
If a polygon has six sides, then it is a hexagon. Hypothesis: Conclusion: b) Another performance will be scheduled if the first one is sold out. A polygon has six sides It is a hexagon The first one is sold out Another performance will be scheduled
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Example 2 Write the statement in if-then form and then identify the hypothesis and conclusion.
Four quarters can be exchanged for a dollar bill. If-then statement: Hypothesis: Conclusion: b) The sum of the measures of two supplementary angles is 180o. If you have four quarters, then it can be exchanged for a dollar bill. You have four quarters It can be exchanged for a dollar bill If two angles are supplementary, then the sum of their measures is 180o. Two angles are supplementary The sum of their measures is 180o.
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Converse Inverse Contrapositive
___________________ is formed by switching the hypothesis (p) and conclusion (q). Inverse ___________________ is formed by negating the hypothesis (p) and conclusion (q). Contrapositive __________________________ is formed by switching and negating p and q.
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Example: Write the converse, inverse, and contrapositive of the following conditional:
Conditional: If Janice lives in Norfolk, then she lives in Virginia. T F Converse: T F Inverse: Contrapositive: T F If Janice lives in Virginia, then she lives in Norfolk. If Janice does not live in Norfolk, then she does not live in Virginia. If Janice does not live in Virginia, then she does not live in Norfolk.
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Counterexample I live in Virginia but I live in Fredericksburg.
A _______________________ is an example that shows that a statement is false. It fits the first part (p), but doesn’t fit the second part (q). Example: Find a counterexample of the statement: If you live in Virginia, then you live in Woodbridge. counterexample: If n is an integer number, then I live in Virginia but I live in Fredericksburg. 1; 12 > 1 counterexample:
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If an angle is a right angle, then it measures 90.
biconditional When a conditional and the converse are both true, the statement is called a ___________________________. These statements are said “p if and only if q” and written Biconditionals can also be written “p iff q”. Example: Write the biconditional as a conditional and a converse. Then, determine if the biconditional is true or false. If it’s false, then give a counterexample. “An angle is a right angle if and only if its measure is 90˚. conditional: __________________________________________ converse: ____________________________________________ If an angle is a right angle, then it measures 90. If an angle is a right angle, then it measures 90.
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Inductive Reasoning is the reasoning that used a number of specific examples to arrive at a conclusion. When you can assume that a pattern will continue, you are using inductive reasoning. Conjecture is the concluding statement reached (educated guess) using inductive reasoning. Example: Movie show times are 8:30am, 9:45am, 11:00am, …… Inductive reasoning is figuring out the pattern. Make a conjecture:______________________________________________ 12:15pm
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2.4 Deductive Reasoning
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Example 1 facts rules definitions properties Inductive reasoning
Deductive Reasoning: Uses ____________, ___________, __________, or ___________ to reach logical conclusions. Example 1 Determine whether each conclusion is based on inductive or deductive reasoning. All of the signature items on the restaurant’s menu shown are noted with a special symbol. Kevin orders a menu item that has this symbol next to it, so he concludes that the menu item that he has ordered is a signature item._________________ Non of the students who ride Raul’s bus own a car. Ebony rides a bus to school, so Raul concludes that Ebony does not own a car.__________________ definitions properties Inductive reasoning Deductive reasoning
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Law of Detachment: If a conditional is true and its hypothesis is true, then its __________________ is true. If p q is a true statement and p is true, then q is true. Example 2 Determine whether each conclusion is valid based on the given information. Given: If three points are noncollinear, they determine a plane. Points A, B, and C lie in plane G. Conclusion: Points A, B and C are noncollinear. b) Given: If a student turns in a permission slip, then the student can go on the field trip. Felipe turned in his permission slip. Conclusion: Felipe can go on the field trip. conclusion Invalid, they could be collinear on plane G Valid
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Example 3 Use deductive reasoning to draw a conclusion from the following statements using the Law of Detachment. A gardener knows that if it rains, the garden will be watered. It is raining. If A is acute, then the mA < 90o. A is acute. If a triangle is equilateral, then it is an acute triangle. The triangle is equilateral. The garden will be watered. mA < 90o. It is an acute triangle.
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p r Example 4 VALID NOT VALID
Law of Syllogism: If two statements (p q and q r) are true statements, then ___________ is a true statement. p r Example 4 Determine whether each conclusion is valid based on the given information. If you do not get enough sleep, then you will be tired. If you are tired, then you will not do well on the test. If you do not get enough sleep, then you will not do well on the test. ________________ b) If Jamal finishes his homework, he will go out with his friends. If Jamal goes out with his friends, he will go to the movies. If Jamal goes out with his friends, then he finishes his homework. ________________ VALID NOT VALID
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Example 5 Use deductive reasoning to draw a conclusion from the following statements using Law of Syllogism. If a number ends in 0, then it is divisible by 10. If a number is divisible by 10, then it is divisible by 5. b) If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. If a number ends in 0, then it is divisible by 5. If a quadrilateral is a square, then it is a rectangle.
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2.5 Postulates and Paragraph Proofs
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Postulates Points, Lines, and Planes
Postulate or axiom: a statement that is accepted as _____ without _______. true proof Postulates Points, Lines, and Planes Words Example 2.1: Through any two points, there is exactly one line. 2.2: Through any three noncollinear points, there is exactly one plane. 2.3: A line contains at least two points. 2.4: A plane contains at least three noncollinear points. 2.5: If two points lie in a plane, then the entire line containing those points lies in that plane. 2.6: If two lines intersect, then their intersection is exactly one point. 2.7: If two planes intersect, then their intersection is a line. Line n is only thru P and R. P R Plane ABC is the only plane thru the noncollinear points A, B, and C. A B C P Q R Line contains points P, Q, and R Plane LBC contains noncollinear points L, B, C, and E. L B C E Points A and B lie in the plane and line m contains A and B, so line m is in the plane. A B m s Lines s and t intersect at point P. P t w Planes F and G intersect in line w. F G
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Points A, B, and C determine a plane.
Example 1 Explain how each statement is true, then state the postulate that can be used to show each statement is true. Points A, B, and C determine a plane. Planes P and Q intersect in line m. Points A, B and C form the three vertices of the roof. Postulate 2.2 states that through any three noncollinear points there is exactly one plane. The edges of the sides of the roof intersect. Planes P and Q of this roof intersect only once in line m. Postulate 2.7: if two planes intersect, then their intersection is a line.
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Example 2 Determine whether each statement is always, sometimes, never true. Explain.
a) Two lines determine a plane. b) Three lines intersect in two points. c) Points A and C determine a line. Always; between any two points there are always at least three noncollinear points. Postulate 2.2 Never; in order for three lines to intersect in two points, two of the lines would have to be the same. Always; through any two points there is exactly one line. Postulate 2.1
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Theorem: a statement or conjecture that has been _______________.
Proof: logical argument in which each statement you make is supported by a statement that is accepted as true. Theorem: a statement or conjecture that has been _______________. Theorem 2.1: Midpoint Theorem If M is the midpoint of AB, then AM MB. proven A M B
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2.6 Algebraic Proof
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Properties of Real Numbers
The following properties are true for any real numbers a, b, and c. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality Distributive Property If a = b, then a + c = b + c. If a = b, then a – c = b – c. If a = b, then ac = bc. If a = b, and c = 0, then a/c = b/c a = a If a = b, then b = a. If a = b, and b = c, then a = c. If a = b, then a may be replaced by b in any equation or expression a(b + c) = ab + ac. Algebraic Proof: proof that is made up of a series of ________________ ______________. algebraic statements.
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Justify each statement.
Example 1 Justify each statement. a) If 5 = y, then y = 5. __________________________ b) 4(x+5) = 4x + 20 ______________________________ Example 2 Prove that if 2x-13 = -5, then x = 4. Write a justification for each step. Statements Reasons 2x-13 = -5 _________________________________ 2x = 8 _________________________________ x = 4 _________________________________ Symmetric Property Distributive Property Given Addition Property Division Property
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Write a two-column proof to verify each conjecture.
statements Two-column proof: contains _________________ and _____________ organized in two columns. reasons Example 3 Write a two-column proof to verify each conjecture. If m∠ABC + m∠CBD = 90, m∠ABC = 3x – 5, and mCBD = x+1, then x = 27. 2 Given: ___________________________ Prove:____________________________ Statements Reasons 1. m∠ABC + m∠CBD = Given m∠ABC = 3x – 5 m∠CBD = x + 1 2.______________________ 2. Substitution Property 3. 6x – 10 + x + 1= _____________________ 4. ______________________ 4. Simplify 5. ______________________ 5. Addition Property 6. x = ______________________ m∠ABC + m∠CBD = 90, m∠ABC = 3x – 5, and mCBD = x+1 2 x = 27 D C B A 2 3x – 5 + x + 1 = 90 2 Multiplication Property 7x – 9 = 180 7x = 189 Division Property
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2.7 Proving Segment Relationships
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Complete the following proof.
Postulate 2.9: Segment Addition Postulate: If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC. Example 1 Complete the following proof. Given: AB DE, B is the midpoint of AC, E is the midpoint of DF Prove: BC EF Statements Reasons _______________________ a. Given _______________________ b. AB = DE b. ________________________ c. ________________________ c. Definition of Midpoint d. BC = DE d. _________________________ e. BC = EF e. _________________________ f. _________________________ f. _________________________ A B C F E D AB DE, B is the midpoint of AC E is the midpoint of DF Definition of Segments ABBC, DE EF Substitution Property Substitution Property BC EF Definition of Segments
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Example 2 Complete the following proof.
Given: BC = DE Prove: AB + DE = AC Proof: Statements Reasons BC = DE 1. __________________ _________________ 2. Segment Addition Postulate AB + DE = AC 3. ______________________ B A E D Given AB + BC = AC Substitution Property
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2.8 Proving Angle Relationships
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Example 1 Given: ______________________ Prove: ______________________
Postulate 2.11: Angle Addition Postulate: D is the interior of ABC if and only if mABD + m DBC = m ABC. B A C D Example 1 If m1 = 23 and mABC = 131, find the measure of 3. Justify each step. Given: ______________________ Prove: ______________________ Statements Reasons _________________________ 1. _____________________ _________________________ 2. Angle Addition Postulate m 3 = _____________________ _________________________ 4. Subtraction Property 1 = 23; ABC = 131 m 3 1 2 3 1 = 23, ABC = 131 Given 1 + 2 + 3 = ABC Substitution Property m3 = 18
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Theorem 2.3: Supplement Theorem: If two angles form a linear pair, then they are supplementary angles. Theorem 2.4: Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. Example 2 6 and 7 form linear pair. If m6 = 3x + 32 and m7 = 5x + 12, find x, m6 and m7. Justify each step. Given:_________________________ Prove: _________________________ Statements Reasons __________________________ 1. ____________________________ __________________________ 2. ____________________________ 3x x + 12 = ____________________________ 8x + 44 = Simplify/Combine like terms __________________________ 5. Subtraction Property x = ____________________________ m 6 = 3x Given __________________________ 8. Substitution __________________________ 9. Simplify m 7= 5x ___________________________ m 7 = 5(17) ___________________________ __________________________ 12. ___________________________ 6 and 7 form a linear pair, m 6 = 3x + 32, m7 = 5x + 12 x, m 6, and m 7 6 and 7 form a linear pair, m 6 = 3x + 32, m7 = 5x + 12 Given 6 + 7 = 180 Supplement Theorem Substitution 8x = 136 Division Property m 6 = 3(17) + 32 m 6 = 83 Given Substitution m 7 = 97 Simplify
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Given: _______________________ Prove: _______________________
Theorem 2.6: Congruent Supplements Theorem: Angles supplementary to the same angle or congruent angles are congruent. Theorem 2.7: Congruent Complements Theorem: Angles complementary to the same angle or to congruent angles are congruent. Example 3 In the figure, 1 and 4 form a linear pair, and m 3 + m 1 = 180. Prove that 3 and 4 are congruent. 4 3 Given: _______________________ Prove: _______________________ Statements _____________________ 1.____________________ _____________________ Given _____________________ Supplement Theorem 1 and 3 are supplementary Def. of Supplementary angles _______________________5.________________________ 1 & 4 form a linear pair, 3 + 1 = 180 3 4 1 & 4 form a linear pair Given 3 + 1 = 180 1 and 4 are supplementary 3 4 Angles Supplementary to the same angle are
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Given 3 4 6x + 16 = 8x 8 = x 3 = 6(8) +2 4 = 8x - 14 4 = 50
Theorem 2.8: Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. Example 4 If 3 and 4 are vertical angles, m3 = 6x + 2 and m4 = 8x – 14, find the m3 and m4. Justify each step. 3 and 4 are vertical angles, m 3 = 6x + 2, m4 = 8x -14 Given: ____________________ Prove: ____________________ Statements 3 and 4 are vertical angles 1. ______________________ _____________________ 2. Vertical Angles Theorem 3. 6x + 2 = 8x – ______________________ 4. _____________________ 4. Addition Property 5. 16 = 2x ______________________ 6. _____________________ 6. Division Property 7. 3 = 6x ______________________ 8. _____________________ 8. Substitution Property 9. 3 = ______________________ 10. ______________________ 10. Given 11. 4 = 8(8) – _____________________ 12. ______________________ 12. Simplify m3 and m4 Given 3 4 (m3 = m 4) Definition of Substitution Property 6x + 16 = 8x Subtraction Property 8 = x Given 3 = 6(8) +2 Simplify 4 = 8x - 14 Substitution Property 4 = 50
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Prove Theorem 2.10 using the figure at the right.
Theorems: Right Angle Theorems 2.9: Perpendicular lines intersect to form four right angles 2.10: All right angles are congruent. 2.11: Perpendicular lines form congruent adjacent angles. 2.12: If two angles are congruent and supplementary, then each angles is a right angle. 2.13: If two congruent angles form a linear pair, then they are right angles. Example 5 Prove Theorem 2.10 using the figure at the right. Given: 1 and 2 are right angles Prove: 1 2 Statements _____________________ 1. Given m 1 = 90, m 2 = _______________________ ______________________ 3. Substitution ______________________ 4. ________________________ 1 2 1 and 2 are right angles Definition of right angle m1 = m2 1 2 Definition of
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