2.2 - subsets · 2/8/2013 · **every set is a subset of itself** the universal set, symbolized by...
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10/3/2014 Section 2.2 1
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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10/3/2014 Section 2.2 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.
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Universal Set
10/3/2014 Section 2.2 3
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10/3/2014 Section 2.2 4
Example 1Subsets
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10/3/2014 Section 2.2 5
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
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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).
10/3/2014 Section 2.2 6
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10/3/2014 Section 2.2 7
Example 3
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10/3/2014 Section 2.2 8
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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10/3/2014 Section 2.2 9
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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10/3/2014 Section 2.2 10
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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10/3/2014 Section 2.2 11
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.
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Practice
10/3/2014 Section 2.2 12
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Practice
10/3/2014 Section 2.2 13
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10/3/2014 Section 2.3 14
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
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Orange
Woodbridge
Bethany
10/3/2014 Section 2.3 15
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10/3/2014 Section 2.3 16
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.
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Example 1Describe a universal set U
10/3/2014 Section 2.3 17
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10/3/2014 Section 2.3 18
Example 2Determining Sets From a Venn Diagram
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10/3/2014 Section 2.3 19
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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10/3/2014 Section 2.3 20
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’.
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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?
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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
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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
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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
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