1.1 Introduction to Sets

Transcription

1 1.1 Introduction to Sets A set is a collection of items. These items are referred to as the elements or members of the set. We usually use upper-case letters, such as A and B, to denote sets. We can write down a set in two ways: Roster notation: Listing all the elements of the set between a pair of curly braces. example, {a, b, c, d}. For Set-builder notation: Defining the set in terms of its properties. For example, the same set {a, b, c, d} can be written as {x x is one of the first four letters in the English alphabet}. Notation Note: 1. Suppose we are given the set S = {a, b, c, d}. We write c S to mean c is an element of the set S. We write f / S to mean f is not an element of the set S. 2. 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. Thus, {1, 2, 3} {1, 2, 3, 4} but {1, 2, 5} {1, 2, 3, 4} because 5 is in the first set but not in the second set. 3. If A B and there is at least one element of B that is not an element of A, then A is a proper subset of B and we write A B. 4. The empty set, witten as or {}, is the set with no elements. For example, {x x > 4 and x < 3} = The empty set is a subset of any set, i.e., A for any set A. When we use the symbols,, or, we should have sets on both sides of the symbol. For example, {c} {a, b, c} makes sense but c {a, b, c} does not. When we use the symbols or /, we should have an element on the left and a set on the right. For example, {c} {a, b, c} does not make sense but c {a, b, c} does. Example: Find all the subsets of {1, 2}. Example: Find all the subsets of {1, d, β}. Note: If a set has n elements, then the number of subsets will be 2 n. The reason for this will become clear in Chapter 2. 1

2 The universal set is the set of all elements being considered and is denoted by U. A Venn diagram is a way of visualizing sets. The universal set is represented by a rectangle and sets are represented as circles inside the universal set. U B A In the Venn diagram above, A is a subset of B, while both are subsets of U. Having set up the notation and terminology, we need to be able to perform operations on sets. We will consider three operations - complement, union, and intersection. Definitions: Given a universal set U and a set A U, the complement of A, written as A c, is the set of all elements that are contained in the universal set U but not in A, i.e., A c = {x x U, x / A} Note: The complement corresponds to negation in the previous chapter. Example: Let U = {1, 2, 3, 4, a, b, c}, A = {1, 2, a}, and B = {x x is an even number in U}. Find A c, B c, U c, and (A c ) c. Complement Rules: If U is a universal set, we must always have U c =, c = U If A is any subset of U, then (A c ) c = A. 2

3 Definition: The union of two sets A and B, written A B, is the set of all elements that belong to A or B, or to both. Thus A B = {x x A or x B or both} Note: Set union corresponds to inclusive disjunction in the previous chapter. Since we are adopting the convention of using the inclusive or, we normally drop the phrase or both, and write A B = {x x A or x B} Example: Let U = {1, 2, 3, 4, a, b, c, d}, A = {1, 2, a, b}, B = {2, 3, a, b, c}, and C = {2, 4, a, c}. Find A B, A A c, A B C, and A (B C). For set union, we have the following properties: A B = B A, A A c = U, and A B C = A (B C) = (A B) C Definition: The intersection of two sets A and B, written as A B, is the set of all elements that belong to both the set A and to the set B. Thus, A B = {x x A and x B} Note: Set intersection corresponds to conjunction in the previous chapter. Example: Let U = {1, 2, 3, 4, a, b, c, d}, A = {1, 2, a, b}, B = {2, 3, a, b, c}, C = {2, 4, a, c}, and D = {3, c, d}. Find A B, A A c, A D, (A B) C, and A (B C). 3

4 Definition: Two sets A and B are disjoint if they have no elements in common, i.e., if A B =. For set intersection, we have the following properties: A B = B A, A A c =, and A B C = (A B) C = A (B C) De Morgan Laws (A B) c = A c B c ( complement of union is intersection of complements ) (A B) c = A c B c ( complement of intersection is union of complements ) Distributive Laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) Example: Find the region that represents (i) the set A c B C c, (ii) the set A (B C) c in the Venn diagram below. U A a d b e f c B g C h Example: Let U be the set of all students at Texas A&M, let F = {x x is a freshman}, and let C = {x x owns a car}. Describe the sets C c, F C c, and (F c C c ) c in words. 4

5 Example: Let U be the set of all students at Texas A&M, let M = {x x is taking a math course this semester} F = {x x plans to go to a football match this semester} C = {x x lives on campus} Use set notation to represent the following sets: (i) The set of students who live on campus and are not taking a math course this semester. (ii) The set of students living on campus who are taking a math course this semester and do not plan to go to a football match this semester. (iii) The set of students who are either not taking a math course this semester or are living on campus but do not plan to go to a football match this semester. (iv) The set of students who plan to go to a football match this semester, and either do not live on campus or are taking a math course this semester. Section 1.1 suggested homework: 1, 2, 3, 5, 7, 11, 13, 17, 21, 25, 28, 33, 37, 41, 43, 45, 49, 51 5