Section 2.2 Subsets Objectives Recognize subsets and use the notation . Determine the number of subsets of a set. Apply concepts of subsets and equivalent sets to infinite sets. 12/7/2018 Section 2.2

Subsets Set A is a subset of set B, expressed as A  B, if every element in set A is also an element in set B. The notation  means that A is not a subset of B. 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. 12/7/2018 Section 2.2

Universal Set   12/7/2018 Section 2.2

Example 1 Subsets Percentage of Tattooed Americans, By Age Group Age Group Percent Tattooed 18-24 13% 25-29 32% 30-39 24% 40-49 14% 50-64 10% 65+ 7% Applying the subset definition to the set of people age 25 -29 in this table: Write in Standard Notation Form 12/7/2018 Section 2.2

Example 1 continued Given: A = {1, 2, 3} B = {1, 2 } Is A a subset of B? No. A  B Is B a subset of A? Yes. B  A 12/7/2018 Section 2.2

Example 2 . Write  the blank to form a true statement. A = { x | x is a person and x lives in San Francisco} B = { x | x is a person and x lives in California} A ____B Solution: A  B A = { 2, 4, 6, 8} B = { 2, 8, 4, 6} 12/7/2018 Section 2.2

Subsets and the Empty Set The Empty Set as a Subset For any set B, Ø  B. For any set B other than the empty set, Ø  B. 12/7/2018 Section 2.2

The Number of Subsets of a Given Set Number of Elements List of All Subsets Number of Subsets { } 1 {a} {a},{ } 2 {a,b} {a,b},{a}, {b},{ } 4 {a,b,c} 3 {a,b,c},{a,b}, {a,c},{ b,c }, {a},{b},{c}, { } 8 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 2n. The number of proper subsets of a set with n elements is 2n – 1. 12/7/2018 Section 2.2

Example 3 Finding the Number of Subsets Find the number of subsets and the number of proper subsets. {a, b, c, d, e } There are 5 elements so there are 25 = 32 subsets { x | x   and 9 ≤ x ≤ 15 } In roster form, we see that there are 7 elements: { 9, 10, 11, 12, 13, 14, 15 } There are 27 = 128 subsets 12/7/2018 Section 2.2

Cardinal Numbers of Infinite Sets Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal number א0 to the set of natural numbers. He used one-to-one correspondences to establish some surprising equivalences between the set of natural numbers and its proper subsets. 12/7/2018 Section 2.2

Practice   12/7/2018 Section 2.2

Practice   12/7/2018 Section 2.2