Thinking Mathematically Seventh Edition Chapter 2 Set Theory

Thinking Mathematically Seventh Edition Chapter 2 Set Theory Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 1

Section 2. 2 Subsets Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 2

Objectives 1. Recognize subsets and use the notation 2. Recognize proper subsets and use the notation 3. Determine the number of subsets of a set. 4. Apply concepts of subsets and equivalent sets to infinite sets. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 3

Subsets Definition of a Subset of a Set A is a subset of set B, expressed as if every element in set A is also an element in set B. means that A is not a subset of B. The notation Set A is not a subset of set B if there is at least one element of set A that is not an element of set B. Every set is a subset of itself. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 4

Example 1: Using the Symbols Is a Sub Set and Is Not a Subset Write in the blank to form a true statement. Set A is a subset of set B. Set A is not a subset of set B. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 5

Proper Subsets Definition of a Proper Subset of a Set A is a proper subset of set B, expressed as if set A is a subset of set B and sets A and B are not equal Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 6

Example 2: Using the Symbols Is a Sub Set and Is Not a Subset Write or both in the blank to form a true statement: a. Solution: b. Solution: Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 7

Subsets and the Empty Set The Empty Set as a Subset 1. For any set B, 2. For any set B other than the empty set, Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 8

The Number of Subsets of a Given Set As we increase the number of elements in the set by one, the number of subsets doubles. The number of subsets of a set with n elements is The number of proper subsets of a set with n elements is Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 9

Example 3: Finding the Number of Subsets and Proper Subsets Find the number of subsets and the number of proper subsets. a. There are 5 elements so there are proper subsets and b. In roster form, we see that there are 7 elements: There are proper subsets and Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 10

The Number of Subsets of Infinite Sets There are natural numbers. subsets It has proper subsets It has Denote is the “smallest” transfinite cardinal number in an infinite hierarchy of different infinities. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 11