Section 2.2 D1Subsets and Set Operations

Math in Our World

Learning Objectives

Define the complement of a set.Find all subsets of a set.Use subset notation.Find the number of subsets for a set.

Universal Set

A universal set, symbolized by U, is the set of all potential elements under consideration for a specific situation.

Once we define a universal set in a given setting, we are restricted to considering only elements from that set. If U = {1, 2, 3, 4, 5, 6, 7, 8}, then the only elements we can use to define other sets in this setting are the integers from 1 to 8.

Complement

The complement of a set A, symbolized A, is the set of elements contained in the universal set that are not in A.

U A

A

This Venn Diagram shows the visual representation of the sets U and A.The complement of a set A is all the things inside the rectangle, U, that are not inside the circle representing set A.

EXAMPLE 1 Finding the Complement of a Set

Let U = {v, w, x, y, z} and A = {w, y, z}. Find A and draw a Venn diagram that illustrates these sets.

U A

w z

yv x

SOLUTIONUsing the list of elements in U, we just have to cross out the ones that are also in A. The elements left over are in A.

U = {v, w, x, y, z}

A = {v, x}

Subsets

If every element of a set A is also an element of a set B, then A is called a subset of B. The symbol is used to designate a subset; in this case, we write A B.

•Every set is a subset of itself. Every element of a set A is of course an element of set A, so A A.

•The empty set is a subset of every set.

EXAMPLE 2 Finding All Subsets of a Set

Find all subsets of A = {American Idol, Survivor}.

SOLUTIONNumber of elements in Subset Subsets with that Number of Elements

2 {American Idol, Survivor}

1 {American Idol}, {Survivor}

0

So the subsets are: {American Idol, Survivor},

{American Idol}, {Survivor},

Proper Subsets

If a set A is a subset of a set B and is not equal to B, then we call A a proper subset of B, and write A B.

The Venn diagram for a proper subset is shown below. In this case, U = {1, 2, 3, 4, 5}, A = {1, 3, 5}, and

B = {1, 3}.U A

1 3

54

2B

EXAMPLE 3 Finding Proper Subsets of a Set

Find all proper subsets of {x, y, z}.

SOLUTIONNumber of elements in Subset Subsets with that Number of

Elements

3 {x, y, z}

2 {x, y}, {x, z}, {y, z}

1 {x}, {y}, {z}

0

So the proper subsets are: {x, y}, {x, z}, {y, z}, {x}, {y}, {z},

We’ll eliminate this one since

it’s equal to the original.

EXAMPLE 4Understanding Subset Notation

State whether each statement is true or false.

(a){1, 3, 5} {1, 3, 5, 7}

(b) {a, b} {a, b}

(c) {x | x N and x > 10} N

(d) {2, 10} {2, 4, 6, 8, 10}

(e) {r, s, t} {t, s, r}

(f ) {Lake Erie, Lake Huron} The set of Great Lakes

. - “not a subset of”

- “not a subset of”

EXAMPLE 4Understanding Subset Notation

SOLUTION(a)All of 1, 3, and 5 are in the second set, so {1, 3, 5} is a subset of

{1, 3, 5, 7}. The statement is true.

(b)Even though {a, b} is a subset of {a, b}, it is not a proper subset, so the statement is false.

(c) Every element in the first set is a natural number, but not all natural numbers are in the set, so that set is a proper subset of the natural numbers. The statement is true.

(d)Both 2 and 10 are elements of the second set, so {2, 10} is a subset, and the statement is false.

(e)The two sets are identical, so {r, s, t} is not a proper subset of {t, s, r}. The statement is true.

(f )Lake Erie and Lake Huron are both Great Lakes, so the statement is true.

EXAMPLE 5Understanding Subset Notation

State whether each statement is true or false.

(a) {5, 10, 15}

(b) {u, v, w, x} {x, w, u}(c) {0} (d)

EXAMPLE 5Understanding Subset Notation

SOLUTION

(a) True: the empty set is a proper subset of every set.

(b) False: v is an element of {u, v, w, x} but not {x, w, u}.

(c) The set on the left has one element, 0. The empty set has no elements, so the statement is false.

(d) The empty set is a subset of itself (as well as every other set), but not a proper subset of itself since it is equal to itself. The statement is false.

Number of Subsets for a Finite Set

If a finite set has n elements, then the set has 2n subsets and 2n – 1 proper subsets.

Number of elements : n 0 1 2 3

Number of subsets : 2n 1 2 4 8

Number of proper subsets : 2n – 1 0 1 3 7

EXAMPLE 6 Finding the Number of Subsets of a Set

Find the number of subsets and proper subsets of the set {1, 3, 5, 7, 9, 11}.

SOLUTION

The set has n = 6 elements, so there are 2n, or 26 = 64, subsets.

Of these, 2n – 1 , or 64 – 1 = 63, are proper.

Classwork

p. 63: 2, 11, 13, 15, 16, 18, 21, 23, 24, 25, 29, 30, 31, 32, 34, 35, 39, 40, 43