1.1 - Introduction to Sets

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1 1.1 - Introduction to Sets Math Blake Boudreaux Department of Mathematics Texas A&M University January 18, 2018 Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

2 Introduction/Definitions The goal of this chapter is to cover probability. To do this, we must first learn the language of sets. Definition A set is a collection of items and is often denoted with the curly brackets {}. Definition These items are called the elements or members of a set. To denote this relationship we typically use the symbol. Example The set containing 1, 2, and 3 is denoted {1, 2, 3}. Also 1 {1, 2, 3}, 2 {1, 2, 3}, and 3 {1, 2, 3}. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

3 More Examples The set containing the letters a, b, c, d, e, f is written {a, b, c, d, e, f }. The set containing,, and is written {,, }. The set containing the previous two sets is written as { {a, b, c, d, e, f }, {,, } }. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

4 Definition It s often necessary to view sets as contained in a larger set. Definition If every element of a set A is also an element of another set B, we say that A is a subset of B and write A B. If A is not a subset of B, we write A B. Example {1, 2, 4} {1, 2, 3, 4} but {1, 2, 3, 4} {1, 2, 4}. Be careful with use of and. Example: Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

5 More Definitions Definition When A B but A B, we can write A B and call A a proper subset of B. Compare this to < vs. Example In the previous example, we noted {1, 2, 4} {1, 2, 3, 4}, but we can say more precisely that {1, 2, 4} {1, 2, 3, 4}. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

6 Definition Sometimes, the curly bracket way of defining sets will not do (e.g. infinite sets). We can also define the set {1, 2, 3, 4} in terms of its characteristics: or {x x is an integer between 1 and 4 (inclusive)} {x x is an integer and 1 x 4}. This is called set-builder notation. Another example is { } {a, b, c, d, e, f } = Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

7 Definition It is also useful to consider sets that contain no elements at all. Definition The empty set, written as or {}, is the set with no elements. By definition the empty set is a subset of every set. The empty set can be used to indicate equations that have no solution: At the other extreme we have the universal set. Unless otherwise stated the letter U is typically denoted as the universal set. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

8 Example Find all subsets of {a, b, c}. Solution: Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

9 Definition Definition Given a universal set U and a set A U, the complement of A, written A c, is the set of all elements that are in U but not in A, that is As a Venn diagram: A c = {x x U, x A} Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

10 Example Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5}. Find A c, B c, U c, c, and (A c ) c in bracket notation. Solution: Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

11 Definition Definition The union of two sets A and B, written A B. is the set of all elements that belong to A, or to B, or to both. Thus A B = {x x A or x B or both}. If the inclusive or is understood, the above can be written as As a Venn diagram: A B = {x x A or x B}. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

12 Example a. Let A = {1, 2, 3, 4} and B = {1, 4, 5, 6}. Find A B and A A c b. Let A = and B = {1, 2}. Find A B. c. Let A = {1, 2, 3} and B = {1, 2, 3}. Find A B. Solution: Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

13 Definition Definition The intersection of two sets A and B, written A B, is the set of all elements that belong to both the set A and to the set B. Thus As a Venn diagram: A B = {x x A and x B}. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

14 Example Find a. {a, b, c, d} {a, c, e}; b. {a, b} {c, d}. Solution: Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

15 Definition Notice in part b of the last example that the intersection between two sets was empty. Definition Two sets A and B are disjoint if they have no elements in common, that is, if A B =. Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

16 (Example) De Morgan Laws Let A and B be sets in the universe U. Prove (A B) c = A c B c. Proof : Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

17 Through similar methods, one can show the following properties of sets A, B, C in the universe U: Law Name of Law A B = B A Commutative law for union A B = B A Commutative law for intersection A (B C) = (A B) C Associative law for union A (B C) = (A B) C Associative law for intersection A (B C) = (A B) (A C) Distributive law for union A (B C) = (A B) (A C) Distributive law for intersection (A B) c = A c B c De Morgan law (A B) c = A c B c De Morgan law Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

18 Practice True or False? a. A b. A A c. 0 = d. {x, y} {x, y, z} e. {x 0 < x < 1} = f. {x 0 < x < 1} = 0. g. {x x(x 1) = 0} = {0, 1} h. {x x < 0} = Blake Boudreaux (Texas A&M University) Introduction to Sets January 18, / 18

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

Review of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special