# Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing them, we will discuss properties that a relation may or may not have. These are defined only in terms of elements that must belong to the subset in certain situations. In situations like this where things are defined abstractly, it pays to learn to do a couple of things. The first is to always test the definitions and see what they mean, that is, to get a feeling for which things satisfy the definitions and which don t, and why. The second is to keep in mind a few concrete examples. These can be used when exploring what the definitions are saying. 3.1 Cartesian Products In the Cartesian plane (or x-y plane), we associate the set of points in the plane with the set of all ordered points (x, y), where x and y are both real numbers. The idea of a Cartesian product of sets replaces R in the description by some other set(s), and drops the geometric interpretation. If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) : (a A) and (b B)}. 1

2 2 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS The following points are worth special attention: The Cartesian product of two sets is a set. The elements of that set are ordered pairs. In each ordered pair, the first component is an element of A, and the second component is an element of B. The points in the x-y plane correspond to the elements of the set R R. For example, if A = {1, 2, 3} and B = {a, b}, then A B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)} and B A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}. Suppose A has m elements and B has n elements. Then, each element of A is the first component of n ordered pairs in A B: one for each element of B. Thus the number of elements in A B equals m n, the number of elements in A times the number of elements in B. This is one way in which the symbol is suggestive notation for the Cartesian product. What should A be? By definition, it is the set of all ordered pairs (a, b) where a A and b. There are no such pairs, as there are no elements b. Hence A =. Similarly, B =. We says that two ordered pairs are equal if the first components are identical and so are the second components. That is, (a, b) = (c, d) if and only if a = c and b = d. This corresponds to (and generalizes) our idea of equality for ordered pairs of real numbers. The example above shows that A B B A in general. This leads to the question of when they are equal. Certainly they are equal if A = B because then A B = A A = B A. They are also equal when A = or B = because, then A B = = B A. We now show that these are the only possibilities where equality can hold. Proposition Let A and B be sets. Then A B = B A if and only if A = B, or A =, or B =. Proof. ( ) We prove the contrapositive. Suppose A and B are non-empty sets such that A B. Then one of them has an element which does not belong to the other. Suppose first that there exists x A such that x B.

3 3.1. CARTESIAN PRODUCTS 3 Since B, the set A B has an ordered pair with first component x, whereas B A has no such ordered pair. Thus A B B A. The argument is similar in the other case, when there exists y B such that y A. ( ). If A = B then A B = A A = B A. If A =, then A B = = B A. The case where B = is similar. The set A (B C) is the set of all ordered pairs where the first component is an element of A, and the second component is an element of B C. That is, the second component is an element of B or an element of C. This is the same collection that would be obtained from the union (A B) (A C), which is made from the union of the set of all ordered pairs where the first component is an element of A and the second component is an element of B, and the set of all ordered pairs where the first component is an element of A, and the second component is an element of C. This is the outline of the proof of the following proposition. Proposition Let A, B and C be sets. Then, A (B C) = (A B) (A C). Proof. (LHS RHS) Let (x, y) A (B C). Then x A and y (B C). That is, y B or y C. This leads to two cases. If y B, then (x, y) A B, and so (x, y) (A B) (A C). If y C, then (x, y) A C, and so (x, y) (A B) (A C). Therefore, A (B C) (A B) (A C). (RHS LHS) Let (x, y) (A B) (A C). Then (x, y) A B or (x, y) A C. This leads to two cases. If (x, y) A B, then x A and y B. Since y B, we have y B C, so (x, y) A (B C). If (x, y) A C, then x A and y C. Since y C, we have y B C, so (x, y) A (B C). Therefore, (A B) (A C) A (B C). The proposition above can also be proved using set builder notation and showing that the two sets are described by logically equivalent expressions. One hint that this is so is in the informal proof outline that precedes the proposition. Another one is in the proof of the proposition: the second part of the proof above is essentially the first part written from bottom to top. Each step is an equivalence rather than just an implication. The same methods can be used to prove the following similar statements:

4 4 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS A (B C) = (A B) (A C); (A B) C = (A C) (B C); (A B) C = (A C) (B C). It is a good exercise to investigate, then prove or disprove as appropriate, similar statements involving the Cartesian product and operations like set difference, A \ B, and symmetric difference, A B. 3.2 Relations Suppose A is the set of all students registered at UVic this term, and B is the set of all courses offered at UVic this term. Then A B is the set of all ordered pairs (s, c), where s is a student registered at UVic this term, and c is a course offered at UVic this term. The set A B represents all possible registrations by a current student in a current course. Certain subsets of A B may be of interest, for example the subset consisting of the pairs where the course is in Science and the student is actually registered in the course, or the subset consisting of the pairs where completion of the course would make the student eligible to receive a degree from the Faculty of Fine Arts. The idea is that relationships between the elements of A and the elements of B can be represented by subsets of A B. A binary relation from a set A to a set B is a subset R A B. A binary relation on a set A is a subset of R A A. The word binary arises because the relation contains pairs of objects. Ternary relations (on A, say) would contain triples of elements, quaternary relations would contain quadruples of elements, and in general n-are relations would contain ordered n-tuples of elements. We will only consider binary relations, so we will drop the adjective binary. When we talk about relations, we mean binary relations. We will focus almost exclusively on relations on a set A. A relation may or may not express a particular type of relationship between its elements. The definition says that a relation is simply a subset. Any subset. It could be that the only relationship between x and y is that

5 3.3. PROPERTIES OF RELATIONS 5 the pair (x, y) belongs to the subset. Subsets like R 1 = and R 2 = A A are perfectly good relations on A. On the other hand, familiar things can be seen as relations. As a sample: Equality between integers is represented by the relation R on Z where (x, y) Z if and only if x = y. Strict inequality between real numbers is represented by the relation S on R where (x, y) S if and only if x > y. The property of being a subset is represented by the relation C on P(U) where (X, Y ) C if and only if X Y. Logical implication between statements p and q is represented by the relation I on the set of all statements (say involving a certain set of Boolean variables) where (p, q) I if and only if p q. Because of these examples, and many others like them involving common mathematical symbols (that express particular relationships), infix notation is used: sometimes we write xry instead of (x, y) R, and say that x is related to y (under R). 3.3 Properties of Relations The relation = on the set of real numbers has the following properties: Every number is equal to itself. If x is equal to y, then y is equal to x. Numbers that are equal to the same number are equal to each other. That is, if x = y and y = z, then x = z. The relation on the set of all propositions (in a finite number of variables) has properties that look strongly similar to these. Every proposition is logically equivalent to itself.

6 6 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS If p is logically to q, then q is logically equivalent to p. Propositions that are logically equivalent to the same proposition are logically equivalent to each other to each other. That is, if p q and q r, then p r. Similarly, the relation on the set of real numbers has the following properties: x x for every x R. If x y and y x, then x = y. If x y and y z, then x z. The relation on the the power set of a set S has similar properties: X X for every X P(S). If X Y and Y X, then X = Y. If X Y and Y Z, then X Z. The relation on the set of all propositions (in a finite number of variables) looks to have the same properties as the previous two, so long as we accept playing the role of =. There is, however, something subtle and beyond the scope of this discussion, going on in the second bullet point because we use instead of =. p p for every proposition x. If p q and q p, then p q. If p q and q r, then p r. It may or may not be clear that the first bullet point in each of the five collections describes the same abstract property. And the same for the third bullet point. The middle bullet point describes the same abstract property in the first two collections and in the first two of the last three, but these two properties are fundamentally different.

7 3.3. PROPERTIES OF RELATIONS 7 The first property in the five collections above is reflexivity. The dictionary defines reflexive as meaning directed back on itself. In a relation, we interpret that as meaning every element is related to itself. Thus, each of the relations described above is reflexive. The second property in the first two collections, but not the last three, is symmetry : if x is related to y, then y is related to x. The third property in all five collections is transitivity : if x is related to y, and y is related to z, then x is related to z. The second property in collection three and four is anti-symmetry : if x is related to y and y is related to x, then x is the same as y. Later, we will see that being anti-symmetric is very different from being not symmetric. We will also get a hint of the origin of the (unfortunate) term anti-symmetric. Formal definitions of these properties follow. It is important to realize that each of these is a property that a particular relation might, or might not, have. A relation R on a set A is: reflexive if (x, x) R for every x A. (Written in infix notation, the condition is xrx for every x A.) symmetric if (y, x) R whenever (x, y) R, for all x, y A. (Written in infix notation, the condition is if xry then yrx, for all x, y A.) transitive if (x, z) R whenever (x, y), (y, z) R, for all x, y, z A. (Written in infix notation, the condition is if xry and yrz, then xrz, for all x, y, z A.) anti-symmetric if x = y whenever (x, y) R and (y, x) R, for all x, y A. (Written in infix notation, the condition is if xry and yrx, then x = y, for all x, y A.) Why are we doing this? Relations that are reflexive, symmetric and transitive behave a lot like equals : they partition the set A into disjoint collections of elements that are the same (equivalent) with respect to whatever property

8 8 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS is used to define the relation. These are called equivalence relations. Relations that are that reflexive, anti-symmetric and transitive behave a lot like less than or equal to in the sense that they imply an ordering of some of the elements of A. To interpret this for the subset relation, think of X Y as reading X precedes or equals Y (there are some pairs of sets for which neither precedes or equals the other). These are called partial orders. What follows are six examples of determining whether or not a relation has the properties defined above. Consider the relation R 1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3)} on the set A = {1, 2, 3}. R 1 is reflexive: A = {1, 2, 3} and (1, 1), (2, 2), (3, 3) R 1. R 1 is not symmetric: (2, 3) R 1 but (3, 2) R 1. R 1 is not anti-symmetric: (1, 2), (2, 1) R 1 but 1 2. R 1 is not transitive: (1, 2), (2, 3) R 1 but (1, 3) R 1. The definition says that a relation is symmetric if, whenever a pair (x, y) is in the relation, so is its reversal (y, x). This means that when x y, either both of (x, y) and (y, x) are in the relation, or neither are. The definition says that a relation is anti-symmetric if, when x y, we never have both of (x, y) and (y, x) in the relation. (The definition is phrased in a way that makes it easy to use in proofs.) This means that x y, either (x, y) is in the relation and (y, x) is not in it, or (y, x) is in the relation and (x, y) is not in it, or neither pair is in it. The only possibility that is not permitted to arise in an anti-symmetric relation is for it to contain both (x, y) and (y, x), where x y. (There is no pair of different elements which are related in a symmetric way.) It is possible for a relation to be both symmetric and anti-symmetric, for example A = {(1, 1), (2, 2)} on the set {1, 2, 3}. The relation R 1 above shows that it is also possible for a relation to be neither symmetric nor antisymmetric. Consider the relation R 2 = on any non-empty set A. R 2 is not reflexive. Since A, there exists x A. The ordered pair (x, x) R 2.

9 3.3. PROPERTIES OF RELATIONS 9 R 2 is symmetric. The implication if (x, y) R 2, then (y, x) R 2 is true because its hypothesis is always false. R 2 is anti-symmetric. The implication if (x, y), (y, x) R 2, then x = y is true because its hypothesis is always false. R 2 is transitive. The implication if (x, y), (y, z) R 2, then (x, z) R 2 is true because its hypothesis is always false. If A =, then what above changes slightly because R 2 is reflexive. Can you explain why? Let R 3 be the subset relation on P(S), the set of all subsets of S = {1, 2, 3, 4}, that is, (X, Y ) R 3 if and only X Y. R 3 is reflexive because X X for every X S (for every X P(S)). R 2 is not symmetric: (, {1}) R 3 because {1} but ({1}, ) R 3 because {1}. R 3 is anti-symmetric. Suppose (X, Y ), (Y, X) R 3. Then X Y and Y X. We proved before that this means X = Y. R 3 is transitive. Suppose (X, Y ), (Y, Z) R 3. Then X Y and Y Z. We proved before that this means X Z, that is (X, Z) R 3. If S =, then what above changes slightly because R 3 is symmetric. Can you explain why? Let R 4 be the relation on N defined by (m, n) R 4 if and only if m n is even. R 4 is reflexive. Let k N. Then k k = 0. Since 0 is even, (k, k) R 4. R 4 is symmetric. Suppose (m, n) R 4. Then m n is even. Since n m = (m n), and the negative of an even number is even, (n, m) R 4. R 4 not anti-symmetric: (1, 3), (3, 1) R 4 but 1 3. R 4 is transitive. Suppose (k, m), (m, n) R 4. Then k m is even, and m n is even. Hence, (k m) + (m n) = k n is even it is the sum of two even numbers. Therefore (k, n) R 4.

10 10 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS Let A be a set with at least two elements, and let R 5 be the relation A A on A. R 5 is reflexive. It contains all possible ordered pairs of elements of A, so it contains (x, x) for every x A. R 5 is symmetric. It contains all possible ordered pairs of elements of A, so it contains (y, x) whenever it contains (x, y). R 5 not anti-symmetric: since A has at least two elements, there exist a, b A such that a b. Since (a, b), (b, a) A A, the statement follows. R 5 is transitive. It contains all possible ordered pairs of elements of A, so it contains (x, z) whenever it contains (x, y) and (y, z). If A has at most one element, then the above changes. In that case, R 5 is anti-symmetric. Can you explain why? Finally, let R 6 be the relation on Z Z defined by (a, b)r 6 (c, d) if and only if a c and b d. Notice that, here, it is pairs of elements that are being related (to each other) under R 6, so technically R 6 is an set of ordered pairs, of which the components are ordered pairs. The infix notation (a, b)r 6 (c, d) is far less cumbersome that writing ((a, b), (c, d)) R 6. R 6 is reflexive. Let (a, b) Z Z. Since a a and b b, we have (a, b)r 6 (a, b). R 6 is not symmetric: (1, 2)R 6 (3, 4) but (3, 4)R 6 (1, 2). R 6 is anti-symmetric. Suppose (a, b)r 6 (c, d) and (c, d)r 6 (a, b). Then a c and b d, and c a and d b. Therefore a = c and b = d, so that (a, b) = (c, d). R 6 is transitive. Suppose (a, b)r 6 (c, d) and (c, d)r 6 (e, f). We want (a, b)r 6 (e, f). Since (a, b)r 6 (c, d), a c and b d. Since (c, d)r 6 (e, f), c e and d f. Therefore a e and b f, so that (a, b)r 6 (e, f). We close this section with a different sort of example. Suppose R is a relation on {1, 2, 3, 4} that is symmetric and transitive. Suppose also that (1, 2), (2, 3), (1, 4) R. What else must be in R?

11 3.3. PROPERTIES OF RELATIONS 11 Since R is symmetric, we must have (2, 1), (3, 2), (4, 1) R. Since R is transitive and (1, 2), (2, 1) R, we must have (1, 1) R. Similarly (2, 2), (3, 3), (4, 4) R. Since (1, 2), (2, 3) R, transitivity implies (1, 3) R. Symmetry gives (3, 1) R. Let s summarize what we have done so far in an array. The rows and columns are indexed by {1, 2, 3, 4}, and the entry in row i and column j is the truth value of the statement (i, j) R (that is, it is 1 if the pair (i, j) R and 0 otherwise Notice that the array is symmetric (in the matrix-theoretic sense): the (i, j)- entry equals the (j, i)-entry. Must (2, 4) be in R? We have (2, 1), (1, 4) R, so (2, 4) R. Thus (4, 2) R by symmetry. What about (3, 4)? We have (3, 1), (1, 4) R, so again the answer is (3, 4) R. Thus (4, 3) R. Therefore R = A A. An array of the type in the previous example rows and columns indexed by elements of A and (i, j)-entry the truth value of the statement (i, j) R denotes a reflexive relation when every entry on the main diagonal equals 1; symmetric relation when it is symmetric about the main diagonal: the (i, j)-entry equals the (j, i)-entry. anti-symmetric relation when there is no i j such that the (i, j)-entry and the (j, i)-entry are both equal to 1. (Entries on the main diagonal don t matter, and it acceptable for the (i, j)-entry and the (j, i)-entry to both equal 0.) It is not easily possible to look at the array and see if the relation is transitive. All of the possibilities need to be checked.

12 12 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS 3.4 Equivalence Relations An equivalence relation on a set A is a relation on A that is reflexive; symmetric; and transitive Relations with these three properties are similar to =. Suppose R is an equivalence relation on A. Instead of saying (x, y) R or x is related to y under R, for the sake of this discussion let s say x is the same as y. The reflexive property then says everything in A is the same as itself. The symmetric property says if x is the same as y, then y is the same as x. And the transitive property says things that are both the same as the same element are the same as each other. Another translation of these statements arises from replacing is the same as by is equivalent to. The following are examples of equivalence relations: logical equivalence on the set of all propositions; the relation R on Z defined by xry if and only if x y is even; the relation T on {0, 1,..., 24} defined by h 1 T h 2 if any only if h 1 hours is the same time as h 2 hours on a 12-hour clock; the relation S on the set of all computer programs defined by p 1 Sp 2 if and only if p 1 computes the same function at p 2 ; the relation E on the set of all algebraic expressions in x defined by p(x) E q(x) if and only if p(x) = q(x) for every real number x. For example, if p(x) = x 2 1 and q(x) = (x + 1)(x 1), then p(x) E q(x). It is a useful exercise to prove that each of these is an equivalence relation. Every equivalence relation carves up (mathematicians would say partitions ) the underlying set into collections (sets) of equivalent things (things that are the same ), where the meaning of equivalent depends on the definition of the relation. In the examples above:

13 3.4. EQUIVALENCE RELATIONS 13 logical equivalence partitions the universe of all statements into collections of statements that mean the same thing, and hence can be freely substituted for each other; R partitions the integers into the even integers and the odd integers; T partitions {0, 1,..., 24} into collections of hours that represent the same time on a 12-hour clock; S partitions the set of all computer programs into collections that do the same thing; E partitions the set of all algebraic expressions into collections that give the same numerical value for every real number x, and hence can be freely substituted for each other when manipulating equations. Each of these collections of equivalent things is an example of what is called an equivalence class. Let R be an equivalence relation on A, and x A. The equivalence class of x is the set [x] = {y : yrx}. Let A be a set. A partition of A is a collection of disjoint, non-empty subsets whose union is A. That is, it is a set of subsets of A such that the empty set is not in the collection; and every element of A belongs to exactly one set in the collection. Each set in the collection is called a cell, or block, or element of the partition. A: For example, if A = {a, b, c, d, e}, then the following are all partitions of {{a}, {b, e}, {c, d}}; {A}; {{a, c, e}, {b, d}}; {{a}, {b}, {c}, {d}, {e}}. None of the following are partitions of A:

14 14 CHAPTER 3. CARTESIAN PRODUCTS AND RELATIONS {{a}, {b, e}, {c, d}, }; {{a, c, e}, {d}}; {{a}, {b}, {c}, {a, d}, {e}}; {a}, {b}, {c}, {d}, {e}. The last example of something that isn t a partition is slippery. It isn t a set, hence it can t be a partition. But this is just a technicality mathematicians frequently write partitions in this way. The point of this example was to make sure you re aware of what happens sometimes, and what is intended. That equivalence relations and partitions are actually two sides of the same coin is the main consequence of the two theorems below. The first theorem says that the collection of equivalence classes is a partition of A (which is consistent with what we observed above). The second theorem says that for any possible partition of A there is an equivalence relation for which the subsets in the collection are exactly the equivalence classes. Theorem Let R be an equivalence relation on A. Then 1. x [x]; 2. if xry then [x] = [y]; and 3. if x is not related to y under R, then [x] [y] =. Proof. The first statement follows because R is reflexive. To see the second statement, suppose xry. If z [x] then (by definition of equivalence classes) zrx. By transitivity, zry. That is z [y]. Therefore [x] [y]. A similar argument proves that [y] [x], so that [x] = [y]. To see the third statement, we proceed by contradiction. Suppose x is not related to y under R, but [x] [y]. Let z [x] [y]. Then zrx and zry. By symmetry, xrz. And then by transitivity, xry, a contradiction. Therefore, [x] [y] =. Part 1 of the above theorem says that the equivalence classes are all nonempty, and parts 2 and 3 together say that every element of X belongs to exactly one equivalence class. Parts 2 and 3 also tell you how to determine

15 3.4. EQUIVALENCE RELATIONS 15 if two equivalence classes are the same: [x] = [y] if and only if x is related to y (equivalently, since R is symmetric, y is related to x). For example, suppose R is the relation on R defined by xry if and only if x rounds to the same integer as y. Then R is an equivalence relation (exercise: prove it). The partition of R induced by R is {[n 0.5, n + 0.5) : n Z}, where each half-open interval [n 0.5, n + 0.5) = {x : n x < n + 0.5}. Among [1], [ 2], [ 3], [2], [e], [π] there are exactly three different equivalence classes because [1] = [ 2]; [ 3] = [2], and [e] = [π]. Theorem Let Π = {X 1, X 2,..., X t } be a partition of a set A. Then the relation R on A defined by xry if and only if x belongs to the same cell of Π as y is an equivalence relation; and Π is the partition of A induced by the set of equivalence classes of R. Proof. The argument that shows R is an equivalence relation is left as an exercise. We argue that Π is the partition of A induced by the set of equivalence classes of R. That is, it must be shown that, for any x A, the equivalence class of x equals the cell of the partition that contains x. Take any x A, and suppose x X i. We need to show that [x] = X i. On the one hand, if y X i then yrx by definition of R. Hence, y [x]. Therefore, X i [x]. On the other hand, if y [x] then yrx. By definition of R, the element y belongs to the same cell of Π as x. That is, y X i. Therefore [x] X i. It now follows that [x] = X i. For example, suppose we want an equivalence relation F on [0, ) for which the partition of R induced by F is {[n, n+1) : n N {0}}. According to the theorem statement, we define xfy if and only if there exists n N {0} such that x, y [n, n + 1). Looking at the definition of F we see that xfy if and only if the integer part of x (the part before the decimal point) is the same as the integer part of y, or equivalently that the greatest integer less than or equal to x (commonly known as the floor of x and denoted x ) is the same as the greatest integer less than or equal to y. In symbols xfy x = y.

### Math 3000 Running Glossary

Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour

Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Week 5: Binary Relations

1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### 2.1.1 Examples of Sets and their Elements

Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### Sets, Relations and Functions

Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

### In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)

Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### 31 is a prime number is a mathematical statement (which happens to be true).

Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### In mathematics there are endless ways that two entities can be related

CHAPTER 11 Relations In mathematics there are endless ways that two entities can be related to each other. Consider the following mathematical statements. 5 < 10 5 5 6 = 30 5 5 80 7 > 4 x y 8 3 a b ( mod

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### Logic, Sets, and Proofs

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Preference Relations and Choice Rules

Preference Relations and Choice Rules Econ 2100, Fall 2015 Lecture 1, 31 August Outline 1 Logistics 2 Binary Relations 1 Definition 2 Properties 3 Upper and Lower Contour Sets 3 Preferences 4 Choice Correspondences

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### (Refer Slide Time 1.50)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Module -2 Lecture #11 Induction Today we shall consider proof

### 2.1 Sets, power sets. Cartesian Products.

Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects. - used to group objects together, - often the objects with similar properties This description of a set (without

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### 2.1 The Algebra of Sets

Chapter 2 Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations

### Page 331, 38.4 Suppose a is a positive integer and p is a prime. Prove that p a if and only if the prime factorization of a contains p.

Page 331, 38.2 Assignment #11 Solutions Factor the following positive integers into primes. a. 25 = 5 2. b. 4200 = 2 3 3 5 2 7. c. 10 10 = 2 10 5 10. d. 19 = 19. e. 1 = 1. Page 331, 38.4 Suppose a is a

### This section demonstrates some different techniques of proving some general statements.

Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)

### This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.

CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers.

Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### Logic will get you from A to B. Imagination will take you everywhere.

Chapter 3 Predicate Logic Logic will get you from A to B. Imagination will take you everywhere. A. Einstein In the previous chapter, we studied propositional logic. This chapter is dedicated to another

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### Classical Analysis I

Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### Solutions to In-Class Problems Week 4, Mon.

Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Matthias Beck Ross Geoghegan. The Art of Proof. Basic Training for Deeper Mathematics

Matthias Beck Ross Geoghegan The Art of Proof Basic Training for Deeper Mathematics ! "#\$\$%&#'!()*+!!,-''!.)-/%)/#0! 1)2#3\$4)0\$!-5!"#\$%)4#\$&*'!! 1)2#3\$4)0\$!-5!"#\$%)4#\$&*#6!7*&)0*)'! 7#0!83#0*&'*-!7\$#\$)!90&:)3'&\$;!

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder,

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;

Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method

### Axiom A.1. Lines, planes and space are sets of points. Space contains all points.

73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly