WEEK 2-3
SETS
2
Introduction to Sets
Introduction to Sets
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A set is an unordered collection of objects, called elements or
members of the set. A set is said to contain its elements. We write
a ∈ A to denote that a is an element of the set A. The notation a
 A denotes that a is not an element of the set A.
Sets are used to group objects together. Often, but not always,
the objects in a set have similar properties.
For instance, all the students who are currently enrolled in your
school make up a set.
Likewise, all the students currently taking a course in mathematics
at any school make up a set.
Introduction to Sets
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Examples: (This way of describing a set is known as the roster method)
1. Cities in the Pakistan: {Lahore, Karachi, Islamabad, … }
2. Sets can contain non-related elements: {3, a, red, Gilgit}

Properties:
1. Order does not matter
{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}
2. Sets do not have duplicate elements
Capital letters (A, B, S…) for sets
Lower-case letters for elements of a set (a, x, y…)
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Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}}
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They are all different
Introduction to Sets
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Easiest way: list all the elements
 A = {1, 2, 3, 4, 5}, Not always feasible to list every element!
May use ellipsis (…) when the general pattern of the elements is obvious:
B = {0, 1, 2, 3, …}
May cause confusion. C = {3, 5, 7, …}. What’s next?
If the set is all odd integers greater than 2, next element is 9
If the set is all prime numbers greater than 2, next element is 11
Introduction to Sets
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D = {x | x is prime ˄ x > 2}
E = {x | x is odd ˄ x > 2}
 The vertical bar means “such that”
 If an element a is a member of a set S, we use the notation a  S, i.e.
4  {1, 2, 3, 4}
 If not, we use the notation a  S, i.e.
 7  {1, 2, 3, 4}
Introduction to Sets
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N = {1, 2, 3, …} is the set of natural numbers
Z = {…, -2, -1, 0, 1, 2, …} is the set of integers
Z+ = {1, 2, 3, …} is the set of positive integers
Q = {p/q | p  Z, q  Z, q ≠ 0} is the set of rational numbers
 Any number that can be expressed as a fraction of two
integers (where the denominator is not zero)
R is the set of real numbers
R+, the set of positive real numbers
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Types of Sets
The Universal Set
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U is the universal set – the set of all of elements (or the
“universe”) from which given any set is drawn

For the set {-2, 0.4, 2}, U would be the set of real numbers
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For the set {0, 1, 2}, U could be the set of whole numbers,
Z, R depending on the context
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For the set of the vowels, U would be all the letters of the
alphabet
The Empty/Null Set
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If a set has zero elements, it is called the empty (or null) set
 Written using the symbol 
 Thus,  = { }
 VERY IMPORTANT
It can be a element of other sets
 { , 1, 2, 3, x } is a valid set
≠{}
 The first is a set of zero elements
 The second is a set of 1 element
 Replace  by { }, and you get: { } ≠ {{ }}
 It’s easier to see that they are not equal that way
Equal Sets
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Two sets are equal if they have the same elements
 {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}
 {1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}
 Two sets are not equal if they do not have the
same elements
• {1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}
We write A = B if A and B are equal sets.
Subset
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If A and B are two sets then set B is said to be subset of set A
if all elements of set of B are also an element of A
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If B = {2, 4, 6}, A = {1, 2, 3, 4, 5, 6, 7} then
B is a subset of A i.e. B  A

If A ={a ,b ,c}, B={b ,c}, C={a ,b ,c ,d}, D={a ,d} then
B⊆A, A⊆C, B⊆C, D⊆C. But D⊈A i.e., D is not a subset of A

For any set S, S  S i.e. S (S  S)
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For any set S,   S i.e. S (  S)
Superset, Proper Subset, Improper Subset
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Is   {1,2,3}?
Does   {1,2,3}?
Is   {,1,2,3}?
Does   {,1,2,3}?
Is {x}  {x}?
Does {x}  {x,{x}}?
Is {x}  {x,{x}}?
Does {x}  {x}?
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Is   {1,2,3}? Yes!
Does   {1,2,3}?
No!
Is   {,1,2,3}?
Yes!
Does   {,1,2,3}? Yes!
Is {x}  {x}? Yes
Does {x}  {x,{x}}? Yes
Is {x}  {x,{x}}? Yes
Does {x}  {x}? No
Set Cardinality
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The cardinality of a set A is the number of
elements in a finite set, written as |A| or n(A)
Examples
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Let 𝑅 = {−2, −3, 0, 1, 2}. Then |𝑅| = 𝑛(𝑅) = 5

|| = 0

Let 𝑆 = {, {𝑎}, {𝑏}, {𝑎, 𝑏}}. Then |𝑆| = 4

Let S be the set of letters in the English alphabet.
Then |S| = 26.
Power Set
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•
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Given 𝑆 = {0, 1}. All the possible subsets of S?
 (as it is a subset of all sets), {0}, {1}, and {0, 1}
The power set of S written as P(S) is the set of all the subsets of S
𝑃(𝑆) = { , {0}, {1}, {0,1} }
•
Note that |𝑆| = 𝑛(𝑆) = 2 and |𝑃(𝑆)| = 𝑛[𝑃(𝑆)] = 4
•
If a set has n elements, then the power set will have 𝟐𝒏 elements.
•
Let 𝑇 = {0, 1, 2}
Then 𝑃(𝑇) = { , {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2} }
Note that |𝑇| = 3 and |𝑃(𝑇)| = 8
•
𝑃() = {  }
Note that || = 0 and |𝑃()| = 1
Equivalent Sets
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Two sets A and B are said to be equivalent if they have the
same cardinality i.e. 𝑛(𝐴) = 𝑛(𝐵). It is denoted by 𝑨~𝑩 .
In general, we can say, two sets are equivalent to each other
if the number of elements in both the sets is equal.
Example: 𝐴 = {𝑎 , 𝑏, 𝑐}, 𝐵 = {1, 2, 3}
Both sets have equal number of elements i.e., 𝑨~𝑩
Singleton Set
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A set with one element is called a singleton set.
A common error is to confuse the empty set ∅ with
the set {∅}, which is a singleton set. The single
element of the set {∅} is the empty set itself!
Sets of Sets
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Sets can contain other sets

S = { {1}, {2}, {3} }
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T = { {1}, {{2}}, {{{3}}} }
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V = { { {1}, {{2}} }, {{{3}}}, { {1}, {{2}}, {{{3}}} } }
• V has only 3 elements!
Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}}

They are all different
Ordered Pair
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An ordered pair is a pair of elements (x,y), written in
a particular order. The ordered pair (x,y) is not the
same ordered pair as (y,x). where x is the xcoordinate and y is the y-coordinate.
Cartesian Product
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The order of elements in a collection is often important.
Because sets are unordered, a different structure is needed to represent ordered collections.
• Note that the Cartesian products A×B and B×A are not equal, unless A=B.
Cartesian Product
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Set Operations
Venn Diagrams
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Represents sets graphically
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The box represents the universal set
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Circles represents the set

Consider set S, which is
the set of all vowels in the
alphabet
Union
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Example
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Let A = {3, 4, 5, 6}, B = {6, 7, 8} and C = {8, 9, 7}.
Then
A ∪ B = {3, 4, 5, 6, 7, 8}
B ∪ C = {6, 7, 8, 9}
A ∪ C = {3, 4, 5, 6, 7, 8, 9}
Intersection
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Example
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Let A = {3, 4, 5, 6}, B = {5, 6, 7}, C = {7, 8, 9}
Then,
A ∩ B = {5, 6},
B ∩ C = {7}
Union vs Intersection
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Union vs Intersection
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Disjoint
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Examples
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Further examples
 {1, 2, 3} and {3, 4, 5} are not disjoint
 {a, b} and {3, 4} are disjoint
 {1, 2} and  are disjoint
• Their intersection is the empty set
  and  are disjoint!
• Because their intersection is the empty set
Difference
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Example
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{1, 2, 3} - {3, 4, 5} = {1, 2}
{a, b} - {3, 4} = {a, b}
{1, 2} -  = {1, 2}
The difference of any set S with the empty set will
be the set itself
Complement
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Let U be a universal set and A be any subset of U, then the elements
of U which are not in A i.e., U - A is the complement of A w.r.t. U is
written as A' = U - A = Ac.
Examples
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let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = { 1, 2, 4, 6, 9}
Then, A' = U - A = {0, 3, 5, 7, 8}
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SET IDENTITES
Example
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Example
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Example
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Example
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Example
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