Sets
Chapter 2
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Sets: A Problem Solving Tool
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Sets
A set is a well-defined collection of objects, called
elements or members of the set.
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Let A = {1, 2, 3, 4}, which means that “A is the set
containing the elements 1, 2, 3, and 4.”
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A “4 is in A” or “4 is an element of the set A”
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A “6 is not an element of A”
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3. Let X = {a, b, x, y}. Fill in the blank with  or  to make
each statement correct.
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a. a___X

b. x___X


c. A___X
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Sets of Numbers Commonly used in Mathematics
N = {x | x is a natural number} = {1, 2, 3, …}
W = {x | x is a whole number} = {0,1, 2, 3, …}
I = {x | x is an integer} = {…, 2, 1, 0,1, 2, 3, …}
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Q = {x | x is a rational number} = {x | x, is of the form
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a/b where a, b are integers and b  0}
Q = {x | x is a rational number} = {x | x has a terminating
or repeating decimal pattern}
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S = {x | x is a irrational number} = {x | x is not rational}
R = {x | x is a real number} = {x | x is the union of the
rational and irrational numbers}
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Describing Sets
1. Verbal or written description
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2. Roster Method or Listing
3. Set-builder notation
The Empty Set
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The symbol { } or Ø represents the empty, or null set.
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NOTE:
{Ø} is not the null set. It is the set containing the null set.
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Describe the following sets using the roster method and set
builder notation.
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The
of1 counting
less than 8.
b. The set of counting numbers between 3 and 8.
c. The set of counting numbers between 6 and 7.
d. The set of counting numbers greater than 3.
a. {1, 2, 3, 4, 5, 6, 7}
{x | x is a counting
number
less than 8}
number
between 3 and 8}
number
between 6 and 7}
number
greater th an 3}
b. {4, 5, 6, 7}
{x | x is a counting
c. { } or φ
{x | x is a counting
d. {4, 5, 6,…}
{x | x is a counting
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Equality of Sets
Two sets A and B are equal, A = B, if they have the
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same
elements
necessarily
listed in the same
order.
State whether the sets A and B are equal.
1. A={2n + 1|n is a counting number}
B={2n – 1|n is a counting number}
2. A = {1,1,2,2,3} , B = {2,1,3}
1. A ≠ B , not equal
2. A = B , equal
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Subsets
The set A is a subset of B, A ⊆ B, if every element of
A is also an element of B.
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Proper Subsets
A set A is said to be a proper subset of B, A ⊂ B, if
A ⊆ B but A B.
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The Universal Set
The universal set, U, is the set of all elements under
discussion.
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List all the subsets and indicate which are the proper subsets
of the given set.
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U1=01{a,
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b. U = {1, 2, 3}
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a. Ø, {a}, {b}, and {a, b}. The first three are proper subsets.
b. Ø
{1}, {2}, {3}
{1, 2}, {1, 3}, {2, 3}
{1, 2, 3}
The first seven are proper subsets.
Note: by definition Ø is a subset of any set.
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Number of Subsets of a Set
n
A
set
of
n
elements
has
2
subsets.
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Refer to the problems on the previous slide:
22 = 4 and 23 = 8.
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Number of Proper Subsets of a Set
A set of n elements has 2n – 1 proper subsets.
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Refer to the problems on the previous slide:
22 1 = 4 1 = 3 and 23 1 = 8 1 = 7.
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Example:
Given: A = { 12, 19, 26,…, 68} Find
a. the number of subsets
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the number
of proper
subsets
To solve this problem you need to determine how many elements are in the set.
Notice the elements in this set are evenly spaced and form what is called an
arithmetic sequence. The next term is found by adding a common difference of
seven in this example. The total number of elements can be found without having
to list all of the elements in this set using the formula An = A1 + (n – 1)d.
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An is the last term in the sequence, A1 is the first term, d is the common difference
and n is the number of terms in the sequence. Substituting the numbers into the
equation and solving for n yields the number of terms in this set.
68 = 12 + (n – 1)(7)
–12 –12
56 = (n – 1)(7)
7
7
8 = n–1
+1
+1
9 = n
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a. Number of subsets = 29 = 512
b. Number of proper subsets = 29 – 1 = 512 – 1 = 511
END
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