Binary Relations Part One

Transcription

1 Binary Relations Part One

2 Outline for Today Binary Relations Reasoning about connections between objects. Equivalence Relations Reasoning about clusters. A Fundamental Theorem How do we know we have the right definition for something?

3 Relationships In CS103, you've seen examples of relationships between sets: A B between numbers: x < y x ₖ y x y between people: p loves q Since these relations focus on connections between two objects, they are called binary relations. The binary here means pertaining to two things, not made of zeros and ones.

4 What exactly is a binary relation?

5

6

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8

9

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11 <

12 10 12 <

13 < < 12

14 <

15 5-2 <

16 < < -2

17 ₃

18 7 10 ₃

19 ₃ ₃ 10

20 6 11 ₃

21 ₃ ₃ 11

22

23 R

24 R a b

25 R a b arb

26 R a b

27 R a b ar b

28 Binary Relations A binary relation over a set A is a predicate R that can be applied to ordered pairs of elements drawn from A. If R is a binary relation over A and it holds for the pair (a, b), we write arb. 3 = 3 5 < 7 Ø N If R is a binary relation over A and it does not hold for the pair (a, b), we write ar b <= 3 N = Ø

29 Properties of Relations Generally speaking, if R is a binary relation over a set A, the order of the operands is significant. For example, 3 < 5, but 5 <= 3. In some relations order is irrelevant; more on that later. Relations are always defined relative to some underlying set. It's not meaningful to ask whether 15, for example, since is defined over sets, not arbitrary objects.

30 Visualizing Relations We can visualize a binary relation R over a set A by drawing the elements of A and drawing an arrow between an element a and an element b if arb is true. Example: the relation a b (meaning a divides b ) over the set {1, 2, 3, 4} looks like this:

31 Visualizing Relations We can visualize a binary relation R over a set A by drawing the elements of A and drawing an arrow between an element a and an element b if arb is true. Example: the relation a b over the set {1, 2, 3, 4} looks like this:

32 Visualizing Relations We can visualize a binary relation R over a set A by drawing the elements of A and drawing an arrow between an element a and an element b if arb is true. Example: the relation a = b over the set {1, 2, 3, 4} looks like this:

33 Visualizing Relations We can visualize a binary relation R over a set A by drawing the elements of A and drawing an arrow between an element a and an element b if arb is true. Example: below is some relation over {1, 2, 3, 4} that's a totally valid relation even though there doesn't appear to be a simple unifying rule

34 Capturing Structure

35 Capturing Structure Binary relations are an excellent way for capturing certain structures that appear in computer science. Today, we'll look at one of them (partitions), and next time we'll see another (prerequisites). Along the way, we'll explore how to write proofs about definitions given in first-order logic.

36 Partitions

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47 Partitions A partition of a set is a way of splitting the set into disjoint, nonempty subsets so that every element belongs to exactly one subset. Two sets are disjoint if their intersection is the empty set; formally, sets S and T are disjoint if S T = Ø. Intuitively, a partition of a set breaks the set apart into smaller pieces. There doesn't have to be any rhyme or reason to what those pieces are, though often there is one.

48 Partitions and Clustering If you have a set of data, you can often learn something from the data by finding a good partition of that data and inspecting the partitions. Usually, the term clustering is used in data analysis rather than partitioning. Interested to learn more? Take CS161 or CS246!

49 What's the connection between partitions and binary relations?

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51 a a

52 a a

53

54 a b

55 a b

56 a b

57 b a

58

59 a b

60 a b

61 a bc

62 b c

63 ab c

64 a c

65 a a a b b a a b b c a c

66 ara arb bra arb brc arc

67 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

68 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

69 Reflexivity Some relations always hold from any element to itself. Examples: x = x for any x. A A for any set A. x ₖ x for any x. Relations of this sort are called reflexive. Formally speaking, a binary relation R over a set A is reflexive if the following first-order statement is true: a A. ara ( Every element is related to itself. )

70 Reflexivity Visualized a A. ara ( Every element is related to itself. )

71 Let Let R be be the the relation relation drawn drawn to to the the left. left. Is Is R reflexive? reflexive? a A. ara ( Every element is related to itself. )

72 a A. ara ( Every element is related to itself. )

73 a A. ara ( Every element is related to itself. ) a

74 a A. ara ( Every element is related to itself. ) a

75 a This This means means that that R is is not not reflexive, reflexive, since since the the first-order logic logic statement given given below below is is not not true. true. a A. ara ( Every element is related to itself. )

76 Is Is reflexive? reflexive? a A. ara ( Every element is related to itself. )

77 Is Is reflexive? reflexive? a??. a a

78 Is Is reflexive? reflexive? Reflexivity is is a property property of of relations, not not individual objects. objects. a??. a a

79 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

80 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

81 Symmetry In some relations, the relative order of the objects doesn't matter. Examples: If x = y, then y = x. If x ₖ y, then y ₖ x. These relations are called symmetric. Formally: a binary relation R over a set A is called symmetric if the following first-order statement is true about R: a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

82 Symmetry Visualized a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

83 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

84 Is This Relation Symmetric? b a a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

85 Is This Relation Symmetric? b a a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

86 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

87 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

88 Is This Relation Symmetric? a b a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

89 Is This Relation Symmetric? a b a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

90 Is This Relation Symmetric? a b a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

91 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

92 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

93 Is This Relation Symmetric? a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

94 Is This Relation Symmetric? a a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

95 Is This Relation Symmetric? a a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

96 Is This Relation Symmetric? a a A. b A. (arb bra) ( If a is related to b, then b is related to a. )

97 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

98 a A. ara a A. b A. (arb bra) a A. b A. c A. (arb brc arc)

99 Transitivity Many relations can be chained together. Examples: If x = y and y = z, then x = z. If R S and S T, then R T. If x ₖ y and y ₖ z, then x ₖ z. These relations are called transitive. A binary relation R over a set A is called transitive if the following first-order statement is true about R: a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

100 Transitivity Visualized a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

101 Is This Relation Transitive? a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

102 Is This Relation Transitive? a b c a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

103 Is This Relation Transitive? a b c a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

104 Is This Relation Transitive? a b c a A. b A. c A. (arb brc arc) ( Whenever a is related to b and b is related to c, we know a is related to c.)

105 Equivalence Relations An equivalence relation is a relation that is reflexive, symmetric and transitive. Some examples: x = y x ₖ y x has the same color as y x has the same shape as y.

106 Binary relations give us a common language to describe common structures.

107 Equivalence Relations Most modern programming languages include some sort of hash table data structure. Java: HashMap C++: std::unordered_map Python: dict If you insert a key/value pair and then try to look up a key, the implementation has to be able to tell whether two keys are equal. Although each language has a different mechanism for specifying this, many languages describe them in similar ways...

108 Equivalence Relations The equals method implements an equivalence relation on non-null object references: It is reflexive: for any non-null reference value x, x.equals(x) should return true. It is symmetric: for any non-null reference values x and y, x.equals(y) should return true if and only if y.equals(x) returns true. It is transitive: for any non-null reference values x, y, and z, if x.equals(y) returns true and y.equals(z) returns true, then x.equals(z) should return true. Java 8 Documentation

109 Equivalence Relations The equals method implements an equivalence relation on non-null object references: It is reflexive: for any non-null reference value x, x.equals(x) should return true. It is symmetric: for any non-null reference values x and y, x.equals(y) should return true if and only if y.equals(x) returns true. It is transitive: for any non-null reference values x, y, and z, if x.equals(y) returns true and y.equals(z) returns true, then x.equals(z) should return true. Java 8 Documentation

110 Equivalence Relations Each unordered associative container is parameterized by Key, by a function object type Hash that meets the Hash requirements ( ) and acts as a hash function for argument values of type Key, and by a binary predicate Pred that induces an equivalence relation on values of type Key. Additionally, unordered_map and unordered_multimap associate an arbitrary mapped type T with the Key. C++14 ISO Spec, /3

111 Equivalence Relations Each unordered associative container is parameterized by Key, by a function object type Hash that meets the Hash requirements ( ) and acts as a hash function for argument values of type Key, and by a binary predicate Pred that induces an equivalence relation on values of type Key. Additionally, unordered_map and unordered_multimap associate an arbitrary mapped type T with the Key. C++14 ISO Spec, /3

112 Time-Out for Announcements!

113 Girls Teaching Girls to Code Exec Team Applications are open now! Want to inspire the next generation of women in tech? Passionate about CS education and outreach? Join the Executive Team of Girls Teaching Girls to Code! Girls Teaching Girls to Code is a program for Stanford students to teach high school girls from around the Bay Area to code. Students learn programming basics, build exciting projects, and develop strong relationships with mentors in the field. We are currently accepting applicants from all majors who are interested in building our corporate relations, coordinating logistics for our events, creating and reviewing curricula, or supporting outreach efforts in the community. This year, we brought together over 200 high school girls and 60 Stanford mentors for a day-long programming experience. Aside from our main event, we also hosted smaller events throughout the year. We received overwhelmingly positive feedback, and we're looking for team leads help us grow GTGTC even more! Learn more about us at and apply at by Friday, October 12, 2018 at 11:59 pm!

114 Problem Set One Solutions We ve just released solutions to Problem Set One, both in hardcopy and online. You need to read over these solutions as soon as possible. Why? Each question is there for a reason. We ve described what it is that we hoped you would have learned when solving those problems. There are lots of different ways of solving these problems. Comparing what you did against our solutions, which are just one possible set of solutions, can help introduce new techniques.

115 Problem Set Two The Problem Set Two checkpoint was due today at 2:30PM. The remaining problems are due Friday at 2:30PM. Have questions? Stop by office hours! Ask on Piazza!

116 Your Questions

117 I m considering an independent contractor job. The contract is confusing- I don t want to get screwed, per se. Where can I get legal advice as an undergrad? You You can can get get free free legal legal advice advice from from the the ASSU ASSU (check (check this this link link for for more more details.) details.) I d I d like like to to hear hear more more about about why why you re you re considering considering this this job. job. If If you you need need the the money money and and they re they re offering offering to to pay pay well, well, or or if if it s it s for for a a project project you re you re genuinely genuinely excited excited about, about, great! great! If If you re you re trying trying to to improve improve your your coding coding skills, skills, make make sure sure that that you re you re actually actually signing signing up up for for something something where where that that will will happen. happen. You You learn learn best best when when you re you re surrounded surrounded by by other other skilled skilled people people who who know know what what they re they re doing. doing. Is Is that that the the case case here? here? If If you re you re doing doing this this to to improve improve your your resume resume are are you you sure sure you you need need to to do do that? that? You You aren t aren t expected expected to to have have past past job job experience experience when when applying applying for for internships internships and and the the like. like.

118 When s a good time and what s the best way to tell my PSet partner I like them? Think Think of of this this as as akin akin to to asking asking a a coworker coworker out out on on a a date. date Keep Keep the the Categorical Categorical Imperative Imperative in in mind mind and and remember remember that that many many people people legitimately legitimately do do not not like like being being hit hit on on in in the the workplace. workplace. As As an an undergrad undergrad there there is is basically basically no no separation separation between between home home and and the the workplace, workplace, but but working working on on problem problem sets sets together together is is way way more more of of the the workplace workplace than than home. home Think Think of of the the position position you you are are putting putting the the other other person person in. in. Your Your ask ask comes comes with with a a real real risk risk of of disrupting disrupting their their work work schedule. schedule. General General advice: advice: give give people people face-saving, face-saving, minimally-disruptive minimally-disruptive ways ways to to say say no. no The The more more important important question question is is do do you you have have a a reason reason to to believe believe the the other other person person enjoys enjoys your your company? company? Because Because if if the the answer answer is is no, no, then then the the answer answer is is never, never, since since otherwise otherwise you re you re weird weird and and awkward. awkward.

119 Back to CS103!

120 Equivalence Relation Proofs Let's suppose you've found a binary relation R over a set A and want to prove that it's an equivalence relation. How exactly would you go about doing this?

121 An Example Relation Consider the binary relation ~ defined over the set Z: a~b if a+b is even Some examples: 0~4 1~9 2~6 5~5 Turns out, this is an equivalence relation! Let's see how to prove it. We We can can binary binary relations relations by by giving giving a a rule, rule, like like this: this: a~b a~b if if some some property property of of a a and and b holds holds This This is is the the general general template template for for defining defining a a relation. relation. Although Although we're we're using using if if rather rather than than iff iff here, here, the the two two above above statements statements are are definitionally definitionally equivalent. equivalent. For For a a variety variety of of reasons, reasons, definitions definitions are are often often introduced introduced with with if if rather rather than than iff. iff. Check Check the the Mathematical Mathematical Vocabulary Vocabulary handout handout for for details. details.

122 What properties must ~ have to be an equivalence relation? Reflexivity Symmetry Transitivity Let's prove each property independently.

123 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

124 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

125 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is What even. is the formal definition of reflexivity? What is the formal definition of reflexivity? To see this, notice that a+a = 2a, so the sum a+a a can be written as a 2k Z. for Z. a some ~ a integer k (namely, a), so a+a is even. Therefore, a~a holds, as Therefore, Therefore, we'll we'll choose choose an an arbitrary arbitrary integer integer a, a, then then required. go go prove prove that that a ~ a. a.

126 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is What even. is the formal definition of reflexivity? What is the formal definition of reflexivity? To see this, notice that a+a = 2a, so the sum a+a a can be written as a 2k Z. for Z. a some ~ a integer k (namely, a), so a+a is even. Therefore, a~a holds, as Therefore, Therefore, we'll we'll choose choose an an arbitrary arbitrary integer integer a, a, then then required. go go prove prove that that a ~ a. a.

127 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is What even. is the formal definition of reflexivity? What is the formal definition of reflexivity? To see this, notice that a+a = 2a, so the sum a+a a can be written as a 2k Z. for Z. a some ~ a integer k (namely, a), so a+a is even. Therefore, a~a holds, as Therefore, Therefore, we'll we'll choose choose an an arbitrary arbitrary integer integer a, a, then then required. go go prove prove that that a ~ a. a.

128 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is What even. is the formal definition of reflexivity? What is the formal definition of reflexivity? To see this, notice that a+a = 2a, so the sum a+a a can be written as a 2k Z. for Z. a some ~ a integer k (namely, a), so a+a is even. Therefore, a~a holds, as Therefore, Therefore, we'll we'll choose choose an an arbitrary arbitrary integer integer a, a, then then required. go go prove prove that that a ~ a. a.

129 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is What even. is the formal definition of reflexivity? What is the formal definition of reflexivity? To see this, notice that a+a = 2a, so the sum a+a a can be written as a 2k Z. for Z. a some ~ a integer k (namely, a), so a+a is even. Therefore, a~a holds, as Therefore, Therefore, we'll we'll choose choose an an arbitrary arbitrary integer integer a, a, then then required. go go prove prove that that a ~ a. a.

130 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

131 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

132 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

133 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

134 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required.

135 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

136 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

137 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because What a+b = b+a, What is the this the formal means formal definition definition of that b+a of symmetry? is symmetry? even. Since b+a is even, we know that b~a, as required. a a Z. Z. b b Z. Z. (a (a ~ b b ~ a) a) Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a and and b where where a ~ b, b, then then prove prove that that b ~ a. a.

138 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because What a+b = b+a, What is the this the formal means formal definition definition of that b+a of symmetry? is symmetry? even. Since b+a is even, we know that b~a, as required. a a Z. Z. b b Z. Z. (a (a ~ b b ~ a) a) Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a and and b where where a ~ b, b, then then prove prove that that b ~ a. a.

139 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because What a+b = b+a, What is the this the formal means formal definition definition of that b+a of symmetry? is symmetry? even. Since b+a is even, we know that b~a, as required. a a Z. Z. b b Z. Z. (a (a ~ b b ~ a) a) Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a and and b where where a ~ b, b, then then prove prove that that b ~ a. a.

140 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because What a+b = b+a, What is the this the formal means formal definition definition of that b+a of symmetry? is symmetry? even. Since b+a is even, we know that b~a, as required. a a Z. Z. b b Z. Z. (a (a ~ b b ~ a) a) Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a and and b where where a ~ b, b, then then prove prove that that b ~ a. a.

141 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because What a+b = b+a, What is the this the formal means formal definition definition of that b+a of symmetry? is symmetry? even. Since b+a is even, we know that b~a, as required. a a Z. Z. b b Z. Z. (a (a ~ b b ~ a) a) Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a and and b where where a ~ b, b, then then prove prove that that b ~ a. a.

142 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

143 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

144 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

145 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

146 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required.

147 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

148 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

149 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that (a+b) + (b+c) = 2k + 2m. Rearranging, a we see that a Z. Z. b b Z. Z. c c Z. Z. (a (a ~ b b ~ c a ~ c) c) a+c + 2b = 2k + 2m, so What What is is the the formal formal definition definition of of transitivity? transitivity? Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a, a, b, b, and and c where where a ~ a+c b and = and 2k b + ~ c, 2m c, then then prove 2b = prove that 2(k+m b). that a ~ c. c. So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

150 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that (a+b) + (b+c) = 2k + 2m. Rearranging, a we see that a Z. Z. b b Z. Z. c c Z. Z. (a (a ~ b b ~ c a ~ c) c) a+c + 2b = 2k + 2m, so What What is is the the formal formal definition definition of of transitivity? transitivity? Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a, a, b, b, and and c where where a ~ a+c b and = and 2k b + ~ c, 2m c, then then prove 2b = prove that 2(k+m b). that a ~ c. c. So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

151 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that (a+b) + (b+c) = 2k + 2m. Rearranging, a we see that a Z. Z. b b Z. Z. c c Z. Z. (a (a ~ b b ~ c a ~ c) c) a+c + 2b = 2k + 2m, so What What is is the the formal formal definition definition of of transitivity? transitivity? Therefore, Therefore, we'll we'll choose choose arbitrary arbitrary integers integers a, a, b, b, and and c where where a ~ a+c b and = and 2k b + ~ c, 2m c, then then prove 2b = prove that 2(k+m b). that a ~ c. c. So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

152 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

153 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a~b and b~c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

154 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

155 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

156 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

157 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

158 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

159 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

160 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

161 An Observation

162 a~b if a+b is even Lemma 1: The binary relation ~ is reflexive. Proof: Consider an arbitrary a Z. We need to prove that a~a. From the definition of the ~ relation, this means that we need to prove that a+a is even. To see this, notice that a+a = 2a, so the sum a+a can be written as 2k for some integer k (namely, a), so a+a is even. Therefore, a~a holds, as required. The The formal formal definition definition of of reflexivity reflexivity is is given given in in first-order first-order logic, logic, but but this this proof proof does does not not contain contain any any first-order first-order logic logic symbols! symbols!

163 a~b if a+b is even Lemma 2: The binary relation ~ is symmetric. Proof: Consider any integers a and b where a~b. We need to show that b~a. Since a~b, we know that a+b is even. Because a+b = b+a, this means that b+a is even. Since b+a is even, we know that b~a, as required. The The formal formal definition definition of of symmetry symmetry is is given given in in first-order first-order logic, logic, but but this this proof proof does does not not contain contain any any first-order first-order logic logic symbols! symbols!

164 a~b if a+b is even Lemma 3: The binary relation ~ is transitive. Proof: Consider arbitrary integers a, b and c where a~b and b~c. We need to prove that a~c, meaning that we need to show that a+c is even. Since a~b and b~c, we know that a+b and b+c are even. This means there are integers k and m where a+b = 2k and b+c = 2m. Notice that Rearranging, we see that so (a+b) + (b+c) = 2k + 2m. a+c + 2b = 2k + 2m, The The formal formal definition definition of of transitivity transitivity is is given given in in first-order first-order logic, logic, but but this this proof proof does does not not contain contain any any first-order first-order logic logic symbols! symbols! a+c = 2k + 2m 2b = 2(k+m b). So there is an integer r, namely k+m b, such that a+c = 2r. Thus a+c is even, so a~c, as required.

165 First-Order Logic and Proofs First-order logic is an excellent tool for giving formal definitions to key terms. While first-order logic guides the structure of proofs, it is exceedingly rare to see first-order logic in written proofs. Follow the example of these proofs: Use the FOL definitions to determine what to assume and what to prove. Write the proof in plain English using the conventions we set up in the first week of the class. Please, please, please, please, please internalize the contents of this slide!

166 Next Time A Fundamental Theorem Why are equivalence relations so useful? Proofs on Relations Proving properties of abstract objects. Strict Orders Representing rankings and requirements. Hasse Diagrams How to draw strict orders.