Chapter 2 Reasoning and Proof Section 2-1 Inductive Reasoning and Conjecture Make Conjectures Inductive reasoning - reasoning that uses a number of specific examples to arrive at a conclusion Conjecture - a concluding statement reached using inductive reasoning Counterexample - an example used to show a conjecture is false Example 1 Write a conjecture to describe the pattern then find the next term. 2, 4, 12, 48, 240 The nth term of the sequence is multiplied by the number n+1, so the next term would be 240 6, or 1440
Example 2 Make a conjecture and provide an example that supports your conjecture For points L, M and N, LM = 20, MN = 6 and LN = 14 L N 14 6 20 L, M and N are collinear M Example 3 The given table shows sales for the first 3 months a store is open. The owner wants to predict sales for the fourth month. Month Sales 1 $500 2 $1500 3 $4500 The sales triple each month, therefore, the predicted sales for the fourth month should be $13,500. Some people may like to use a graph with this as well, but it is not required. Example 4 Use the data in the table to write a counterexample to the following statement Unemployment is highest in the counties with the greatest population County Population Unemployment Armstrong 2163 3.7% Cameron 371825 7.2% El Paso 713126 7.0% Hopkins 33201 4.3% Maverick 50436 11.3% Mitchell 9402 6.1% Maverick county has the highest unemployment rate, but not the highest population Section 2-2 Logic
Truth Statement - a sentence that is either true or false. Statements are often represented with a p or a q. Truth value - the status of the truth of a statement, i.e. true or false Negation - a statement that has the opposite meaning, usually formed by adding the word not to the original statement. The negation of p is not p or ~p More Truth Compound statement - a statement formed by combining two statements using the words and or or Conjunction - a compound statement formed using the word and. A conjunction is true only when both statements are true. A conjunction of the statements p and q is noted as p and q or p q Still Truth Disjunction - a compound statement formed using the word or. A disjunction is true if either of the statements is true. A disjunction of the statements p and q would be denoted as p or q or p q. Truth table - a method of organizing the truth values of statements The symbol is known as a logical and operator. The symbol is known as a logical or operator. Example 1 Use the following to write each conjunction and determine the truth value. p: one foot is 14 inches q: September has 30 days r: a plane is defined by 3 non collinear points p q ~p r one foot has fourteen inches and September has 30 days FALSE one foot is not 14 inches and a plane is defined by 3 non collinear points TRUE
Example 2 Use the statements to construct compound statements and determine the truth values p: AB is the proper notation for segment AB q: Inches are an english unit r: 10 is a prime number p q q r ~p r AB is the proper notation for segment AB or inches are an english unit inches are and english unit or 10 is a prime number AB is not the proper notation for segment AB or 10 is a prime number TRUE TRUE TRUE Example 3 Construct a truth table for the statements p ~q and p (q ~r). p q ~q p ~q T T F F T F T T F T F F F F T F p q r ~r q ~r p (q ~r) T T T F T T T T F T T T T F T F F F T F F T T T F T T F T F F T F T T F F F T F F F F F F T T F Venn Diagrams Illustration Venn diagram - a diagram using intersecting shapes to show compound statements. Intersections (overlapping) in the shapes shows conjunctions while the simple combining of the shapes shows disjunction p q p ~q ~p q p q p q
Example 4 How many play all 3 sports? How many play football or baseball? 136 How many play football or basketball but 84 not baseball? 9 29 Baseball 25 Football 43 13 9 28 Basketball 17 Section 2-3 Conditional Statements If-Then Conditional statement - a statement that can be written in if then form containing a hypothesis and conclusion If-Then Statement - a statement written in the form If p then q where it can be written in the form p q read if p then q or p implied q More Definitions Hypothesis - the statement immediately following if in and if-then statement Conclusion - the statement immediately following then in an if-then statement It is possible to write conditional statements without the use of the words if and then and in such cases the hypothesis and conclusion must be identified
Example 1 Identify the hypothesis and conclusion of the following: If a polygon has 6 sides, then it is a hexagon H: a polygon has 6 sides C: it is a hexagon Example 2 Identify the hypothesis and conclusion and write the conditional in if-then form for the following: A whale is a mammal H: An animal is a whale C: It is a mammal If an animal is a whale, then it is a mammal Truth Values in Conditionals Example 3 p q p q T T T T F F F T T F F T In a conditional statement, the conditional is only false if the hypothesis is true and the conclusion is false When the hypothesis is false, the conditional is always considered true, regardless of the conclusion Determine the truth value: If you subtract a whole number from a whole number the result is a whole number. FALSE If last month was February, this month is march. TRUE When a rectangle has an obtuse angle it is a parallelogram. FALSE
Related Conditionals Related conditional - other statements that are based on the original conditional statement. Converse - the hypothesis and conclusion of are exchanged Inverse - the hypothesis and conclusion are negated Contrapositive - the hypothesis and conclusion are both negated and exchanged Clarification Original statement Inverse Converse Contrapositive If an angle has a measure of 90 then it is a right angle If it is a right angle then an angle has a measure of 90 If an angle does not have a measure of 90 then it is not a right angle If it is not a right angle then an angle does not have a measure of 90 p q q p ~p ~q ~q ~p Logically equivalent - statements with the same truth values. Logical Equivalence A conditional and its contrapositive will be logically equivalent, that is to say, either both will be true or both will be false. An inverse and a converse of a conditional will be logically equivalent. A conditional and its converse may, or may not be logically equivalent. Example 4 Write the inverse, converse and contrapositive of the following: If we annoy Mr. Prusik, then he will make our lives difficult If we don t annoy Mr. Prusik, then he won t make our lives difficult If he will make our lives difficult, then we annoy Mr. Prusik If he will not make our lives difficult, then we don t annoy Mr. Prusik
Biconditional Statements Biconditional - a conditional statement such that the original statement and the converse are both true. The conjunction of a conditional and its converse ( p q) ( q p) ( p q) p iff q IFF? IFF is the common abbreviation for the phrase if and only if, which is only used in a biconditional. Section 2-4 Deductive Reasoning Reasoning Deductive reasoning - using facts, rules, definitions or properties to reach logical conclusions from given statements Valid - logically correct. Valid methods are used to prove conjecture with deductive reasoning Law of Detachment - If p q is a true statement and p is true, then q is true Inductive vs. Deductive Deductive reasoning is called a top down approach to reasoning, it means that you take general information and use it to make specific claims, or use facts you have learned. Inductive reasoning is called a bottom up approach, it means that you use specific information (including patterns) to make general claims.
Example 1 Inductive or Deductive? In Mike s town, April is the month that has had the most rain for the last 5 years. He thinks april will have the most rain this year. Inductive, it uses a observed pattern to make a claim Mike s science book says that if it is cloudy at night, it will not be as cold in the morning as it would be if there were no clouds. Mike knows it will be cloudy tonight and believes it will not be cold tomorrow morning. Deductive, it uses specific facts learned Example 2 Determine if the stated conclusion is valid. If it is invalid, state a reason. If a number is divisible by 4 then the number is divisible by 2. 12 is divisible by 4 Conclusion: 12 is divisible by 2 VALID If Sam stays up late, he will be tired. Sam is tired. Conclusion: Sam stayed up late INVALID there are other reasons why someone could be tired Example 3 Is the conclusion valid based on the given information. Given: If a triangle is equilateral, then it is acute. The triangle is equilateral Conclusion: The triangle is acute Triangles The conclusion is valid Acute Law of Syllogism Law of Syllogism - If p q and q r are true statements, then p r is a true statement This is kind of like logical leapfrog, allowing you to jump over q to make a statement about the direct relationship between p and r. Equilateral
Example 4 Draw a valid conclusion 1. If Jim finishes his homework, he will go out with his friends. 2. If Jim goes out with his friends, he will go to the movies Given: Jim finished his homework p: Jim finishes his homework Jim goes to the q: Jim goes out with his friends movies is a valid r: Jim goes to the movies conclusion by the law of syllogism Example 5 Draw a valid conclusion if possible Given: If it snows more than 5 inches, school will be closed. It snows 7 inches. p: it snows more than 5 inches q: school will be closed Since it snows 7 inches, the hypothesis p is true. By the law of detachment a valid conclusion is school is closed Section 2-5 Postulates and Paragraph Proofs Points, Lines and Planes, Redux Postulate - a statement that is accepted as true without being proven. It is also called an axiom Some postulates are so special they have names, but most postulates are numbered in text books today solely so you know where to find them.
Postulates Postulate 2.1 - Through any two points there is exactly one line Postulate 2.2 - Through any three non collinear points there is exactly one plane Postulate 2.3 - A line contains at least two points Postulate 2.4 - A plane contains at least 3 noncollinear points Postulates again Postulate 2.5 - If two points line in a plane than the entire line containing those points lies in that plane Postulate 2.6 - If two lines intersect, their intersection is exactly one point Postulate 2.7 - If two planes intersect, their intersection is exactly one line Example 1 State the postulate(s) the illustrates why the statement is true: Points A and C lie in plane ABC and on line m. Line m lies entirely in the given plane. Postulate 2.5! Points D and E determine a line Postulate 2.1 m A B E D C Example 2 Is the statement always, sometimes or never true. Explain. If a plane contains a line, and that line contains a point, then the given plane contains that same point. ALWAYS! A line contains 3 noncollinear points. NEVER
Paragraph Proofs The Proof Process Proof - a logical argument in which each statement you make is supported by a statement that is accepted as true Theorem - a conjecture that has been proven and is acceptable to be used as a reason to justify statements in a proof When we come across theorems for the first time, we must prove them, then we can start using them 1. List the given information and draw a diagram if possible 2. State the theorem or conjecture to be proven 3. Create a deductive argument by forming a logical chain of statements linking the given to what you are trying to prove 4. Justify each statement with a reason, including definitions, postulates, properties and already proven theorems 5. State what you have just proven We do not refer to theorems and postulates by number, that is only for organizational reasons. More Proofs Paragraph proof - a written paragraph to explain why a conjecture for a given situation is true. Also called an informal proof. This does not imply that it is any less valid than any other type of proof. Example 3 Given: M is the midpoint of XY Prove: XM MY X M We know that XM = MY because of the definition of a midpoint. We also know that XM MY because of the definition of congruence. Y
Midpoints Midpoint Theorem (2.1) - If M is the midpoint of AB then AM MB. A Just like postulates, sometimes theorems and corollaries are special enough to have a name. M We just proved this theorem on the previous slide. B Section 2-6 Algebraic Proof Properties of Real Numbers Proofs Property 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 Propoerty of Equality Substitution Property of Equality Distributive Property Illustration 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 then a c = b c,c 0 a = a If a = b then b = a If a = b and b = c then a = c If a = b, a can be replaced by b a b + c ( ) = ab + ac Algebraic proof - a proof that is a series of algebraic statements. Algebraic properties are used as justification (reasons) for each line of work in the proof Two column proof - a proof used to prove conjectures and theorems that contains statements and reasons used to justify statements organized into two columns. Also called a formal proof
Example 1 Solve, with justification for each step 2( 5 3a) 4( a + 7) = 92 10 6a 4a 28 = 92 10a 18 = 92 10a = 110 a = 11 Original Equation Distributive Property Substitution Property (=) Addition Property (=) Division Property (=) Example 2 The distance an object travels (d) in a time period (t) is given by the equation:! d = 20t + 5 Verify that the time it takes (t) for an object to travel a given distance (d) is modeled by the following:! t = d 5 20 Write a two column proof Example 2 continued Geometric Proof In a classic 2 column proof, the first line of your proof is your given, and the last line is what you were told to prove (verify). Statement d = 20t + 5 Reasons Given d 5 = 20t Subtraction prop (=) d 5 20 = t Div. Prop. (=) t = d 5 20 Symmetric prop. (=) In geometry the properties listed before are often used, but we have to be careful with notation when dealing with things like angles and line segments Property Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Illustration a + b = b + a ab = ba ( a + b) + c = a + b + c ( a i b)i c = a i( b i c) ( )
Given! Prove m A = 90 Example 3 A B,m B = 2m C,m C = 45 Example 3 Statement Reasons A B,m B = 2m C,m C = 45 Given m A = m B Def. of s m A = 2m C Trans. prop (=) m A = 2 45 ( ) Substitution prop (=) m A = 90 Substitution prop (=) Postulates Section 2-7 Proving Segment Relationships Ruler postulate (2.8) - The points on any line or segment can be put into one to one correspondence with real numbers Segment addition postulate (2.9) - If A, B and C are collinear, then B is between A and C if and only if AB + BC = AC A B C
Example 1 Given: AB CD Prove: AC BD A B C D Given: AB CD Prove: AC BD A B C D Statement AB CD AB = CD AB + BC = AC,CD + BC = BD Reasons Given Def. of seg. Seg. add. postulate AB + BC = BD Substitution prop (=) AC = BD Substitution prop (=) AC BD Def. of seg. Segment Congruence Like properties of equality we saw last section, there are also properties of congruence we use in our proofs. Property Reflexive Property of Congruence Illustration AB AB Some books call these properties theorems, some do not. Example 2 Given: WY = YZ,YZ XZ, XZ WX Prove: WX XY Y Z Symmetric Property of Congruence If AB CD then CD AB W X Transitive Property of Congruence If AB CD and CD EF then AB EF
Given: WY = YZ,YZ XZ, XZ WX Prove: WX XY Y Z Statement WY = YZ,YZ XZ, XZ WX WY YZ WX YZ W Reasons Given Def of seg Trans prop ( ) X Section 2-8 Proving Angle Relationships WX XY Trans prop ( ) Postulates Protractor postulate (2.10) - Given any angle, the measure can be put into one to one correspondence with real numbers between 0 and 180. Angle addition postulate (2.11) - D is in the interior of ABC if and only if m ABD + m DBC = m ABC Illustration B C A D m BAC + m CAD = m BAD
Example 1 If m 1 = 23 and m ABC = 131, find the measure of 3. m 1+ m 2 + m 3 = m ABC A 1 23+ 90 + m 3 = 131 B 2 3 C 113+ m 3 = 131 m 3 = 18 A 1 2 B Example 1a 3 C Justify each step m 1+ m 2 + m 3 = m ABC add. pos 23+ 90 + m 3 = 131 Substitution prop (=) 113+ m 3 = 131 Substitution prop (=) m 3 = 18 Subtraction prop (=) Theorems Angle Congruence Supplement theorem (2.3) - If two angles for a linear pair they they are supplementary angles. 1 2 m 1+ m 2 = 180 Property Reflexive Property of Congruence Symmetric Property of Congruence Illustration 1 1 If 1 2 then 2 1 Complement Theorem (2.4) - If the noncommon sides of two adjacent angles form a right angle then the angles are complementary angles 1 2 m 1+ m 2 = 90 Transitive Property of Congruence Again, some books call these properties theorems, some do not. If 1 2 and 2 3 then 1 3
Theorems Illustrations Congruent supplements theorem (2.6) - Angles supplementary to the same angle or to congruent angles are congruent. Congruent complements theorem (2.7) - Angles complementary to the same angle or to congruent angles are congruent. 2 3 5 6 1 4 If m 1+ m 2 = 180 and m 1+ m 3 = 180 then 2 3 If m 4 + m 5 = 90 and m 4 + m 6 = 90 then 5 6 Given: 2 6 Prove: 4 8 Example 2 Statement 2 6 1& 2 Are supplementary 5 & 6 Are supplementary 1& 4 Are supplementary 5 & 8 Are supplementary 1 2 4 3 5 6 8 7 Reasons Given Supplements thm Supplements thm Given: 2 6 Prove: 4 8 Statement 2 6 1& 2 Are supplementary 5 & 6 Are supplementary 1& 4 Are supplementary 5 & 8 Are supplementary 4 2, 6 8 4 6 4 8 1 2 4 3 5 6 8 7 Reasons Given Supplements thm Supplements thm Congruent suppl. thm Trans prop ( ) Trans prop ( )
Example 3 Given: Statement ABE and DBC ABE and DBC are right angles are rt s Prove: ABC & DBE are comp ABD EBC CBE & DBE are comp A D ABD EBC E Reasons Given Def of comp s congruent comp thm Theorem Vertical angle theorem (2.8) - If two angles are vertical angles they are congruent. 7 4 6 5 4 6 7 5 B C Given: 2 6 Prove: 4 8 Example 4 Statement 1 2 4 3 2 6 2 & 4 are vert s 6 & 8 are vert s 2 4, 6 8 4 2 4 8 5 6 8 7 Reasons Given Def of vert s Vert s are Sym prop ( ) Trans prop ( ) Right Angle Theorems Theorem 2.9 - Perpendicular lines form 4 right angles. Theorem 2.10 - All right angles are congruent Theorem 2.11 - Perpendicular lines form congruent adjacent angles Theorem 2.12 - If two angles are congruent and supplementary they are right angles Theorem 2.13 - If two angles form a linear pair, then they are right angles