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Math 166, Fall 2015, Robert
Williams
1.1- Introduction to Sets
Notation
A set is a collection of items called the elements of the set. Sets are usually
denoted with capital letters and their elements are usually denoted by lowercase
letters. There are two ways to write a set:
• To write a set in roster notation, we list the elements inside of curly
braces. X = {2, 3, 5}
• In set-builder notation, we describe the elements of the set by rules
that only they satisfy. X = {x|x is prime and x < 6}
Sets care about
, but they do not care about
or
.
For example, {x, 1, apple} = {1, x, apple, 1, apple, 1}.
The symbol ∈ is read as
Example 1 Practice using set notation by doing the following:
• Write V = {x|x is a vowel in the English alphabet} in roster notation
• Use set-builder notation to write A = {1, 3, 5, 7, 9}
• Circle the correct statement:
a) 11 ∈ A
11 ∈
/A
b) u ∈ V
u∈
/V
c) u ⊆ V
{u} ⊆ V
d) {1, 3} ∈ A
{1, 3} ⊆ A
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Math 166, Fall 2015, Robert
Williams
If A and B are sets, we call A a subset of B if
. We may write this as A ⊆ B.
We say that A is a proper subset of B if
.
We may write this as A ⊂ B.
A special set that comes up often is the empty set. This is the set with no
elements and is written either as ∅ or {}. The empty set is a subset of every set!
Example 2 Fill in the blank to make a true statement.
∅ = {x|
}
Example 3 Find all subsets of {x, y}.
If A is a set of n elements, then A has 2n subsets.
How many subsets does {1, 2, 3, 4} have?
How many proper subsets does {1, 2, 3, 4} have?
The universal set is the set of
. We denote
the universal set by U .
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Math 166, Fall 2015, Robert
Williams
Set Operations
There are three major set operations that we are interested in, which we will
illustrate using Venn diagrams.
The complement of A, denoted by AC , is
.
U
A
The union of two sets A and B, denoted by A∪B, is
.
U
A
B
The intersection of two sets A and B, denote by A∩B, is
.
U
A
B
Two sets A and B are called disjoint if
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Math 166, Fall 2015, Robert
Williams
Example 4 Let U = {1, 2, 3, 4, 5, 6, 7} be the universal set. Let A = {1, 2, 4, 6},
B = {3, 4}, C = {x|x is an odd number between 0 and 8}. Find the following
sets:
• CC =
• A∪B =
• A∩B∩C =
Useful Properties of Set Operations:
1. ∅C = U and U C = ∅
2. (AC )C = A
3. A ∪ AC = U
4. A ∩ AC = ∅
5. De Morgan’s Laws:
(a) (A ∪ B)C = AC ∩ B C
(b) (A ∩ B)C = AC ∪ B C
Verify the second De Morgan Law using Venn diagrams:
A
B
A
B
A
B
A
B
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Math 166, Fall 2015, Robert
Williams
Example 5 Let U be the universal set of all students at Texas A&M. Consider
the subsets
H = {x|x is currently taking a history class }
M = {x|x is currently taking a math class }
B = {x|x is currently taking a business class }
Write an expression using set operations for each of the following:
• Students who are taking a math class and a business class
• Students who are taking a history class but not a math class
• Students who are taking a history class, or taking a business class, or not
taking a math class
What do the following sets represent in words:
• HC ∩ B
• M ∪H
• H ∩ (M ∪ B)
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