Transcript Document

The Basic
Concepts of Set
Theory
Chapter 1
Set Operations and Cartesian Products
Intersection of Sets
The intersection of sets A and B, written A B,
is the set of elements common to both A and
B, or
A B  {x | x  A and x  B}.
Example: Intersection of Sets
Find each intersection.
a) {1,3,5, 7,9} {1, 2,3, 4,5, 6}
b) {2, 4, 6} 
Solution
a) {1, 3, 5}
b) 
Union of Sets
The union of sets A and B, written A B,
is the set of elements belonging to either of
the sets, or
A B  {x | x  A and x  B}.
Example: Union of Sets
Find each union.
a) {1,3,5, 7,9} {1, 2,3, 4,5, 6}
b) {2, 4, 6} 
Solution
a) {1, 2,3, 4,5, 6, 7,9}
b) {2, 4, 6}
Difference of Sets
The difference of sets A and B, written A – B,
is the set of elements belonging to set A and
not to set B, or
A  B  {x | x  A and x  B}.
Example: Difference of Sets
Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, h},
B = {c, e, g}, and C = {a, c, d, g, e}.
Find each set.
a) A  B
b)  B  A C
Solution
a) {a, b, h}
b) {g} {b, f, h} = {b, f, g, h}
Universal Set and Subsets
• The Universal Set denoted by U is the set of all
possible elements used in a problem.
• When every element of one set is also an element of
another set, we say the first set is a subset.
• Example A={1, 2, 3, 4, 5} and B={2, 3}
We say that B is a subset of A. The notation we use is
B  A.
• Let S={1,2,3}, list all the subsets of S.
• The subsets of S are  , {1}, {2}, {3}, {1,2}, {1,3},
{2,3}, {1,2,3}.
The Empty Set
• The empty set is a special set. It contains no
elements. It is usually denoted as { } or
.
• The empty set is always considered a subset of
any set.
• Is this set {0} empty?
• It is not empty! It contains the element zero.
Cardinal Number
• The Cardinal Number of a set is the number
of elements in the set and is denoted by n(A).
• Let A={2,4,6,8,10}, then n(A)=5.
• The Cardinal Number formula for the union of
two sets is
n(A U B)=n(A) + n(B) – n(A∩B).
• The Cardinal number formula for the
complement of a set is n(A) + n(A’)=n(U).
Ordered Pairs
In the ordered pair (a, b), a is called the
first component and b is called the second
component. In general (a, b)  (b, a ).
Two ordered pairs are equal provided that
their first components are equal and their
second components are equal.
Cartesian Product of Sets
The Cartesian product of sets A and B,
written, A  B, is
A  B  {(a, b) | a  A and b  B}.
Example: Finding Cartesian Products
Let A = {a, b}, B = {1, 2, 3}
Find each set.
a) A  B
b) B  B
Solution
a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),
(3, 1), (3, 2), (3, 3)}
Cardinal Number of a Cartesian Product
If n(A) = a and n(B) = b, then
n  A  B   n( B  A)
 n( A)  n( B)  n( B)  n( A)
 ab  ba
Example: Finding Cardinal Numbers of
Cartesian Products
If n(A) = 12 and n(B) = 7, then find
n  A  B  and n  B  A .
Solution
n  A  B   n( B  A)
 n( A)  n( B)  n( B)  n( A)
 7 12  84
Venn Diagrams of Set Operations
A B
A
A B
B
A
U
U
A
A
U
B
A B
A
A
U
B
Example: Shading Venn Diagrams to
Represent Sets
Draw a Venn Diagram to represent the set
A B.
A
U
B
Example: Shading Venn Diagrams to
Represent Sets
Draw a Venn Diagram to represent the set
 A
B C.
B
A
U
C