Chap2_SETS_PDF.pdf

GEC131 College Mathematic
SETS

➢A set is a collection of well defined distinct
objects.
➢ Objects in the collection are called elements
of the set.
Introduction to Sets

Examples - set
•The collection of all books in our
library is a set.
Each book is an element of the set.
• The collection of all students is a
set.
Each student is an element of the
set.

Examples - set
The collection of counting numbers is a set.
Each counting number is an element of the
set.
The collection of pens in your briefcase is a set.
Each pen in your briefcase is an element of
the set.

Sets Notation
The simplest set notation used to represent the
elements of a set is the curly brackets { }.
An example of a set is A = {a, b, c, d}.
Here the set is represented with a capital letter, and
its elements are denoted by small letters.

Example – roster method
A variation of the simple roster method uses the
ellipsis ( … ) when the pattern is obvious and the set
is large.
{1, 3, 5, 7, … , 9017} is the set of odd
counting numbers less than or equal to 9017.
We call a finite set
{1, 2, 3, 4,… } is the set of all counting
numbers. We call an infinite set.

Example – set builder notation
• {x | x < 6 and x is a counting number} is the set of
all counting numbers less than 6.
• Note this is the same set as {1,2,3,4,5}

Notation – if an element of
If x is an element of the set A, we write this as
x  A. x  A means x is not an element of A.
If A = {3, 17, 2 } then
3  A, 17  A, 2  A and 5  A.
If A = { x | x is a prime number } then
5  A, and 6  A.

Definition
The set with no elements is called the
empty set or void or the null set and is
designated with the symbol .

Examples – empty set
⚫ The set of even prime numbers
greater than 2 is the empty set.
⚫ The set {x | x < 3 and x > 5} is the
empty set.

Definition - subset
The set A is a subset of the set B if every
element of A is an element of B.
• If A is a subset of B we write
A  B to designate that relationship.
• If A is not a subset of B we write
A  B to designate that relationship.

Example - subset
The set A = {3, 5, 7} is not a subset of
set B = {1, 4, 5, 7, 9}
because 3 is an element of A but is not an
element of B.
The empty set is a subset of every set,
because every element of the empty set
is an element of every other set.

Example - subset
The set
A = {1, 2, 3, 4, 5, 6} is a subset of the set
B = {x | x < 7 and x is a counting number}
Notice also that B is a subset of A because every
element of B is an element of A.

Proper Subset
 A is a proper subset of B, written as A  B,
when A is a subset of B and x  B and x 
A (every element A is in B but there is
at least one element B not in A)
 When A is a proper subset of B, A  B
 Eg:
The set of all men is a proper subset of the
set of all people.
{1, 3}  {1, 2, 3, 4}
{1, 2, 3, 4}  {1, 2, 3, 4}

Example - equality
The sets
A = {3, 4, 6} and B = {6, 3, 4} are
equal because A  B and B  A.
The definition of equality of sets shows that
the order in which elements are written
does not affect the set.

Example - equality
The sets A = {2, 5} and
B = {2, 5, 7} are not equal
We would write A ≠ B.
Note that A  B.

Definition - intersection
The intersection of two sets A and B is
the set containing those elements which
are elements of A and elements of B.
We write A  B

If A = {3, 4, 6} and
B = { 1, 2, 3, 4, 5, 6}
then
A  B = {3, 4, 6}.
Example - Intersection

Definition - union
The union of two sets A and B is the set
containing those elements which are
elements of A or elements of B without
duplication
We write A  B

Example - Union
If A = {3, 4, 6} and
B = { 1, 2, 3, 5, 6} then
A  B = {1, 2, 3, 4, 5, 6}.

Algebraic Properties
Union and intersection are
commutative operations.
A  B = B  A
A ∩ B = B ∩ A

Union and intersection are
associative operations.
(A  B)  C = A  (B  C)
(A ∩ B) ∩ C = B ∩ (A ∩ C)

Two distributive laws are true.
A ∩ ( B  C )= (A ∩ B)  (A ∩ C)
A  ( B ∩ C )= (A  B) ∩ (A  C)

A few other elementary properties of
intersection and union.
A   =A A ∩  = 
A  A = A A ∩ A = A

Universal Sets
 The universal set is the set of all
things pertinent to a given discussion
and is designated by the symbol U
Example:
U = {all students at SPUIC}
Some Subsets:
A = {all IAB students}
B = {freshmen students}
C = {sophomore students}

The Complement of a Set
A
The shaded region represents the
complement of the set A
Ac

Set Complement
~A or
A′
 “A complement,” or “not A” is the set
of all elements not in A.
*What the others have that you don’t*

Complements
o If A is a subset of the universal set U,
then the complement of A is the set
o Note: ;

=
 c
A
A
 
c
A x U x A
=  
U
A
A c
=


Cardinality of Sets
 If A is a finite set, then cardinality of A,
denoted by n(A) or |A|, the number of
elements A contains
 Eg: A = {x, y, z}. n(A) = 3
 We see that
A  B  |A|  |B|, and
A  B  |A|  |B|.

Example
 For U = {1, 2, 3, 4, 5}, A = {1, 2},
and B = {1, 2}, we see that A is a subset of
B (that is, A  B), but it is not a proper
subset of B (or, A  B).
 A  B  |A|  |B|, 2  2 so it is true that A
 B
 A  B  |A|  |B|, 2  2 false so it is false
that A  B.

Disjoint Set
 Two sets are disjoint, if they have no elements in
common
 A and B are disjoint, A  B = 
 Eg: Are A = {1, 3, 5} and B = {2, 4, 6} disjoint?
Yes. {1, 3, 5}  {2, 4, 6} = 
 Sets A1, A2, …, An are mutually disjoint, if no two sets Ai
and Aj with distinct subscripts have any elements in
common
 Ai  Aj =  whenever i  j
 Eg: Are B1 = {2, 4, 6}, B2 = {3, 7}, B3 = {4, 5}
mutually disjoint?
No. B1 and B3 both contain 4.

Power Set
 Power set of A, denoted by P(A), is the
set of all subsets of A
 Theorem: If A  B, then P(A)  P(B)
 Theorem: If set X has n elements, then
P(X) has 2n elements
 Eg: P({x, y}) = {, {x}, {y}, {x, y}}

Power Set
 Eg: Let C = {1, 2, 3, 4}. What is
the power set of C?
P(C) = {, {1}, {2}, {3}, {4}, {1,
2}, {1, 3}, {1, 4}, {2, 3}, {2, 4},
{3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3,
4},{2, 3, 4}, C}

Example
 In a class of 50 college freshmen, 30 are studying C++,
25 are studying Java, and 10 are studying both
languages. How many freshmen are studying either
computer language?
We let U be the class of 50 freshmen, A is the subset of
those students studying C++, and B is the subset of
those studying Java.
n(A) = 30, n(B) = 25, n(A  B) = 10
n(A  B) = n(A) + n(B) - n(A  B)
= 30 + 25 – 10
= 45

Example
 Let U = {1, 2, 3, 4, 5, 6, x, y, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}
Then |U| = 11.
(a) If A = {1, 2, 3, 4} then |A| = 4 and
A  U A  U A  U
{A}  U {A}  U {A}  U
(b) Let B = {5, 6, x, y, A} = {5, 6, x, y, {1, 2, 3, 4}}. Then |B| =
5, not 8. And
A  B {A}  B {A}  B
{A}  B A  B (that is, A is not a subset of B)
A  B (that is, A is not a proper subset of B).

Based on the diagram, what is the total number of
students who did participate in volleyball?
15 9 11
6
5 4
12
Volleyball
Example

Examples
{1,2,3}
A = {3,4,5,6}
B =
{3}
A B
 =
{1,2,3,4,5,6}
A B
 =

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