Review of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16

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1 Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16

2 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special Symbols 4 Operations on Sets 5 Results About Sets 6 Exercises. Philippe B. Laval (KSU) Sets Current Semester 2 / 16

3 Introduction This section is only intended as a quick review of material the reader needs to know for this class. For a more thorough treatment of sets, the reader is invited to read books on set theory. Philippe B. Laval (KSU) Sets Current Semester 3 / 16

4 Definitions, Notations and Examples Definitions Definition A set is a well-defined, unordered collection of distinct objects. Well-defined means that given an object and a set, we can tell if the object is in the set. The objects in a set are called its elements. If x is an element of a set A, we write x A. This is read "x is an element of A". A set can be finite or infinite. Philippe B. Laval (KSU) Sets Current Semester 4 / 16

5 Definitions, Notations and Examples Definitions Definition A set is a well-defined, unordered collection of distinct objects. Well-defined means that given an object and a set, we can tell if the object is in the set. The objects in a set are called its elements. If x is an element of a set A, we write x A. This is read "x is an element of A". A set can be finite or infinite. Sets are usually named using a capital letter such as A, B,... The words sets, collection, family are all synonymous. Philippe B. Laval (KSU) Sets Current Semester 4 / 16

6 Definitions, Notations and Examples Definitions Definition A set is a well-defined, unordered collection of distinct objects. Well-defined means that given an object and a set, we can tell if the object is in the set. The objects in a set are called its elements. If x is an element of a set A, we write x A. This is read "x is an element of A". A set can be finite or infinite. Sets are usually named using a capital letter such as A, B,... The words sets, collection, family are all synonymous. The order in which the elements are listed is not relevant. In other words, two sets are equal if and only if they contain the same elements, regardless of the order in which they appear. Philippe B. Laval (KSU) Sets Current Semester 4 / 16

7 Definitions, Notations and Examples Notations and Examples There are different ways of representing a set. We list two of them here. For examples, see the notes. 1 Roster method: We can list all the elements of the set, or, we list enough elements until a pattern is established, then we add.... In both cases, the elements are surrounded by curly brackets. Philippe B. Laval (KSU) Sets Current Semester 5 / 16

8 Definitions, Notations and Examples Notations and Examples There are different ways of representing a set. We list two of them here. For examples, see the notes. 1 Roster method: We can list all the elements of the set, or, we list enough elements until a pattern is established, then we add.... In both cases, the elements are surrounded by curly brackets. 2 Set builder method: We give a rule or a condition to be satisfied in order to belong to a set. If P (x) denotes the condition to be satisfied, we can define a set by writing {x : P (x)}. This means the set of x s such that P (x) is true. Since this is real analysis, we will assume that the x s come from R. If this is not the case, we must specify the set they come from. If they come from a set A, then we write {x A : P (x)}. Philippe B. Laval (KSU) Sets Current Semester 5 / 16

9 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

10 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. / means is not a member of. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

11 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. / means is not a member of. means "is included in but not equal to" or is a proper subset of. It is used between two sets. When we write A B, we mean that A is contained in B, the two are not equal. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

12 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. / means is not a member of. means "is included in but not equal to" or is a proper subset of. It is used between two sets. When we write A B, we mean that A is contained in B, the two are not equal. means "is included in and could be equal to" or is a subset of ; it is used between two sets. When we write A B, it is understood that A is contained in B and could be equal to B. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

13 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. / means is not a member of. means "is included in but not equal to" or is a proper subset of. It is used between two sets. When we write A B, we mean that A is contained in B, the two are not equal. means "is included in and could be equal to" or is a subset of ; it is used between two sets. When we write A B, it is understood that A is contained in B and could be equal to B. denotes the empty set. It is also called the void set or the null set. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

14 Special Symbols Symbols There are some special symbols associated with sets. Some of these symbols include: means is a member of. When we write 2 A, we mean that the element 2 is a member of the set called A. / means is not a member of. means "is included in but not equal to" or is a proper subset of. It is used between two sets. When we write A B, we mean that A is contained in B, the two are not equal. means "is included in and could be equal to" or is a subset of ; it is used between two sets. When we write A B, it is understood that A is contained in B and could be equal to B. denotes the empty set. It is also called the void set or the null set. If A is a set, we denote A the cardinality of A that is the number of elements of A. Philippe B. Laval (KSU) Sets Current Semester 6 / 16

15 Special Symbols Remarks Let A and B be two sets. 1 To show A B, one must show that if we pick an arbitrary element x in A then x is also in B. Philippe B. Laval (KSU) Sets Current Semester 7 / 16

16 Special Symbols Remarks Let A and B be two sets. 1 To show A B, one must show that if we pick an arbitrary element x in A then x is also in B. 2 To show A B one must show that A B and that B has at least one element which is not in A. Philippe B. Laval (KSU) Sets Current Semester 7 / 16

17 Special Symbols Remarks Let A and B be two sets. 1 To show A B, one must show that if we pick an arbitrary element x in A then x is also in B. 2 To show A B one must show that A B and that B has at least one element which is not in A. 3 To show A = B, one must show A B and B A. Philippe B. Laval (KSU) Sets Current Semester 7 / 16

18 Special Symbols Remarks Let A and B be two sets. 1 To show A B, one must show that if we pick an arbitrary element x in A then x is also in B. 2 To show A B one must show that A B and that B has at least one element which is not in A. 3 To show A = B, one must show A B and B A. 4 It should be clear that A for every set A. Philippe B. Laval (KSU) Sets Current Semester 7 / 16

19 Special Symbols Examples Some sets which are used often have a special name. N = {1, 2, 3,...} represents the set of natural numbers or positive integers. Philippe B. Laval (KSU) Sets Current Semester 8 / 16

20 Special Symbols Examples Some sets which are used often have a special name. N = {1, 2, 3,...} represents the set of natural numbers or positive integers. Z = {..., 2, 1, 0, 1, 2,...} represents the set of integers. Philippe B. Laval (KSU) Sets Current Semester 8 / 16

21 Special Symbols Examples Some sets which are used often have a special name. N = {1, 2, 3,...} represents the set of natural numbers or positive integers. Z = {..., 2, 1, 0, 1, 2,...} represents the set of integers. { m } Q = n : m, n Z and n 0 represents the set of rational numbers. Philippe B. Laval (KSU) Sets Current Semester 8 / 16

22 Special Symbols Examples Some sets which are used often have a special name. N = {1, 2, 3,...} represents the set of natural numbers or positive integers. Z = {..., 2, 1, 0, 1, 2,...} represents the set of integers. { m } Q = n : m, n Z and n 0 represents the set of rational numbers. R represents the set of real numbers. Philippe B. Laval (KSU) Sets Current Semester 8 / 16

23 Special Symbols Examples Some sets which are used often have a special name. N = {1, 2, 3,...} represents the set of natural numbers or positive integers. Z = {..., 2, 1, 0, 1, 2,...} represents the set of integers. { m } Q = n : m, n Z and n 0 represents the set of rational numbers. R represents the set of real numbers. If A is any set, then the power set of A, denoted P (A), is the set of all subsets of A (including and A). For example if A = {1, 2, 3} then P (A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} It can be shown that if A has n elements, then P (A) has 2 n elements that is P (A) = 2 n. Philippe B. Laval (KSU) Sets Current Semester 8 / 16

24 Operations on Sets There are four operations defined on sets you should know about. They are: 1 The Union, denoted Philippe B. Laval (KSU) Sets Current Semester 9 / 16

25 Operations on Sets There are four operations defined on sets you should know about. They are: 1 The Union, denoted 2 The Intersection, denoted Philippe B. Laval (KSU) Sets Current Semester 9 / 16

26 Operations on Sets There are four operations defined on sets you should know about. They are: 1 The Union, denoted 2 The Intersection, denoted 3 The difference or relative complement, denoted or \ Philippe B. Laval (KSU) Sets Current Semester 9 / 16

27 Operations on Sets There are four operations defined on sets you should know about. They are: 1 The Union, denoted 2 The Intersection, denoted 3 The difference or relative complement, denoted or \ 4 The Cartesian product, denoted. Philippe B. Laval (KSU) Sets Current Semester 9 / 16

28 Operations on Sets Union and Intersection The union of A and B, denoted A B is defined by: A B = {x : x A or x B} that is it consists of the elements which are either in A or in B. Philippe B. Laval (KSU) Sets Current Semester 10 / 16

29 Operations on Sets Union and Intersection The union of A and B, denoted A B is defined by: A B = {x : x A or x B} that is it consists of the elements which are either in A or in B. The intersection of A and B, denoted A B is defined by: A B = {x : x A and x B} that is it consists of the elements which are in both A and B. Philippe B. Laval (KSU) Sets Current Semester 10 / 16

30 Operations on Sets Difference or relative complement The relative complement of B with respect to A, denoted A \ B or A B (we will use A \ B as A B also means {a b : a A and b B}) is defined by: A \ B = {x : x A and x / B} that is it consists of the elements of A which are not also in B. Philippe B. Laval (KSU) Sets Current Semester 11 / 16

31 Operations on Sets Difference or relative complement The relative complement of B with respect to A, denoted A \ B or A B (we will use A \ B as A B also means {a b : a A and b B}) is defined by: A \ B = {x : x A and x / B} that is it consists of the elements of A which are not also in B. If X is the universal set (or "the world") then instead of writing X \ A or X A, we write A C. This set is called the complement of A. It is understood that it is the complement of A in X. For example, in these notes, R is "the world", the set where all our elements come from, unless specified otherwise. So, if we write A C, it will mean R A. Philippe B. Laval (KSU) Sets Current Semester 11 / 16

32 Operations on Sets Difference or relative complement The relative complement of B with respect to A, denoted A \ B or A B (we will use A \ B as A B also means {a b : a A and b B}) is defined by: A \ B = {x : x A and x / B} that is it consists of the elements of A which are not also in B. If X is the universal set (or "the world") then instead of writing X \ A or X A, we write A C. This set is called the complement of A. It is understood that it is the complement of A in X. For example, in these notes, R is "the world", the set where all our elements come from, unless specified otherwise. So, if we write A C, it will mean R A. If A and B are two sets, A \ B B \ A in general. Philippe B. Laval (KSU) Sets Current Semester 11 / 16

33 Operations on Sets Cartesian Product The Cartesian product of A and B, denoted A B is defined by: A B = {(x, y) : x A and y B} This is called the Cartesian product of A and B. It is the set of all ordered pairs whose first element comes from A and second element from B. Philippe B. Laval (KSU) Sets Current Semester 12 / 16

34 Operations on Sets Cartesian Product The Cartesian product of A and B, denoted A B is defined by: A B = {(x, y) : x A and y B} This is called the Cartesian product of A and B. It is the set of all ordered pairs whose first element comes from A and second element from B. Because it is a set of ordered pairs, if A and B are different sets, A B B A. Philippe B. Laval (KSU) Sets Current Semester 12 / 16

35 Operations on Sets Cartesian Product The Cartesian product of A and B, denoted A B is defined by: A B = {(x, y) : x A and y B} This is called the Cartesian product of A and B. It is the set of all ordered pairs whose first element comes from A and second element from B. Because it is a set of ordered pairs, if A and B are different sets, A B B A. We can take the cartesian product of more than two sets. If A 1, A 2,..., A n are n sets then A 1 A 2... A n = {(a 1, a 2,..., a n ) where a i A i for i = 1, 2,..., n} (a 1, a 2,..., a n ) is called an n-tuple. Philippe B. Laval (KSU) Sets Current Semester 12 / 16

36 Results About Sets Some Remarks and Additional Definitions Let us make a few remarks before we state and prove some important results. The standard technique to prove that two sets A and B are equal (that is A = B) is to prove that A B and B A. Philippe B. Laval (KSU) Sets Current Semester 13 / 16

37 Results About Sets Some Remarks and Additional Definitions Let us make a few remarks before we state and prove some important results. The standard technique to prove that two sets A and B are equal (that is A = B) is to prove that A B and B A. Often, when proving some result about the elements of a set A, we start by picking an arbitrary element of that set by using a statement like "Let x A...". However, this statement makes sense only if A has elements, in other words A. If we do not know that fact, we will have to consider the special case A = as part of the proof. Philippe B. Laval (KSU) Sets Current Semester 13 / 16

38 Results About Sets Some Remarks and Additional Definitions Let us make a few remarks before we state and prove some important results. The standard technique to prove that two sets A and B are equal (that is A = B) is to prove that A B and B A. Often, when proving some result about the elements of a set A, we start by picking an arbitrary element of that set by using a statement like "Let x A...". However, this statement makes sense only if A has elements, in other words A. If we do not know that fact, we will have to consider the special case A = as part of the proof. It should be clear to the reader that if A = or B = then A B =. Philippe B. Laval (KSU) Sets Current Semester 13 / 16

39 Results About Sets Some Remarks and Additional Definitions Let us make a few remarks before we state and prove some important results. The standard technique to prove that two sets A and B are equal (that is A = B) is to prove that A B and B A. Often, when proving some result about the elements of a set A, we start by picking an arbitrary element of that set by using a statement like "Let x A...". However, this statement makes sense only if A has elements, in other words A. If we do not know that fact, we will have to consider the special case A = as part of the proof. It should be clear to the reader that if A = or B = then A B =. Two nonempty sets A and B are said to be disjoint if they do not intersect that is if A B =. Philippe B. Laval (KSU) Sets Current Semester 13 / 16

40 Results About Sets Basic results Theorem Let A, B, and C be three sets. Then, the following properties are satisfied: 1 A, A A = A, A A = A 2 A =, A = A 3 Commutative law. A B = B A and A B = B A 4 Associative law. A (B C) = (A B) C and A (B C) = (A B) C 5 Distributive laws. A (B C) = (A B) (A C) and A (B C) = (A B) (A C) 6 If in addition, A B then A B = B and A B = A Theorem A B and A \ B are disjoint sets. Furthermore, A = (A B) (A \ B). Philippe B. Laval (KSU) Sets Current Semester 14 / 16

41 Results About Sets De Morgan s Laws Theorem Let A, B, and C be sets. Then the following holds: 1 A \ (B C) = (A \ B) (A \ C) 2 A \ (B C) = (A \ B) (A \ C) If we assume that both A and B are subsets of a fixed set X and we wish to express the relative complement with respect to X, then the theorem can be written Theorem Let A and B be sets. Then the following holds: 1 (A B) c = A c B c 2 (A B) c = A c B c Philippe B. Laval (KSU) Sets Current Semester 15 / 16

42 Exercises See the problems at the end of my notes on Review of Sets. Philippe B. Laval (KSU) Sets Current Semester 16 / 16

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