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Section 2.2Subsets

Objectives1. Recognize subsets and use the notation .

2. Determine the number of subsets of a set.

3. Apply concepts of subsets and equivalent sets to infinite sets.

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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.

Universal Set

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Example 1Subsets

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Example 2

• 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

Proper Subsets

• Set A is a proper subset of set B, expressed as A B, if set A is a subset of set B and sets A and B are not equal ( A ≠ B).

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Example 3

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Subsets and the Empty Set

• The Empty Set as a Subset

1. For any set B, Ø B.

2. For any set B other than the empty set, Ø B.

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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 2n.• The number of proper subsets of a set with n elements is

2n – 1.

Set Number of Elements

List of All Subsets Number of Subsets

{ } 0

{a} 1

{a,b} 2

{a,b,c}3

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Example 3Finding the Number of Subsets

Find the number of subsets and the number of proper subsets.

a. {a, b, c, d, e } There are 5 elements so there are 25 = 32 subsets and 25-1

= 31 proper subsets.

b. { 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 and 27-1 = 127 proper subsets.

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Cardinal Numbers of Infinite Sets

Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal number0א 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.

Practice

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Practice

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Section 2.3Venn Diagrams and Subsets

Objectives1. Understand the meaning of a universal set.2. Understand the basic ideas of a Venn diagram.3. Use Venn diagrams to visualize relationships between the

Universal set and a subset. 4. Find the complement of a set

Orange

Woodbridge

Bethany

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Universal Sets and Venn Diagrams

• The universal set is a general set that contains all elements under discussion.

• John Venn (1843 – 1923) created Venndiagrams to show the visual relationship among sets.

• Universal set is represented by a rectangle

• Subsets within the universal set are depicted by circles, or sometimes ovals or other shapes.

Example 1Describe a universal set U

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Example 2Determining Sets From a Venn Diagram

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The Complement of a Set

• The complement of set A, symbolized by A’ is the set of all elements in the universal set that are not in A.

• The shaded region represents the complement of set A. This region lies outside the circle.

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Example 3Finding a Set’s Complement

• Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 4, 7 }. Find A’.

Name: Period: _____ Date: _______________

2.2 Subsets

Set A is a subset of Set B, expressed as if

The notation means

**Every set is a subset of itself**

The Universal Set, symbolized by is

Describe a universal set that includes all elements in sets A and B:

{ }

{ }

A possible Universal Set for sets A and B is

Example #1

Determine whether set A is a subset of set B. Use the correct notation!

a) { }

{ }

b) { } { }

c) { | }

{ | }

Example #2

{ } { }

Is A a subset of B?

Is B a subset of A?

Set A is a proper subset of set B, expressed as , if

Example #3

Write or the blank to form a true statement.

{ | }

{ | }

{ }

{ }

Subsets and the Empty Set

For any set B

For any set B other than the empty set,

The Number of Subsets of a Given Set

Set Number of Elements

List of All Subsets Number of Subsets

{ } 0

{a} 1

{a,b} 2

{a,b,c} 3

The number of subsets of a set elements is

The number of proper subsets of a set elements is

The universal set is

Example #1

Describe a universal set U that describes all the elements in the given sets.

{ }

{ }

U=

Fill in the Venn Diagram below with sets A and B

Example #2

Use the Venn diagram to determine each of the following sets:

a)

b)

c) The set elements in that are not in

The complement of set A, symbolized by , is

Example #3

Let { } and { }. Find .

2.2 - Subsets.pdf2.2 Notes Sheet