Transcript Set Operations - University of New Mexico

Set Operations
Chapter 2 Sec 3
Union
 What
does the word mean to
you?
 What
does it mean in
mathematics?
Definition
 The
union of sets A and B,
written A  B , is the set of
elements that are members of
either A or B(or both).
 Using set-builder notation,
A  B  {x:x is a member of A or
x is a member of B}.
The
union of more than
two sets is the set of all
elements belonging to at
least one of the sets.
Example
 Find
the union of the following
pair of sets.
◦ A={1,3,5,8,9} and B={2,4,6,7,8,}
Solution
 A B
 The
= {1,2,3,4,…,9}
solution in the final answer
we list the elements in order and
did not list duplicate elements
because doing so does not affect
set equality.
Intersection
 The
essential idea of intersection
is a region that is common to
both sets.
Definition
 The
intersection of sets A and B,
written A  B , is the set of
elements common to both A and
B.
 Using set-builder notation,
A  B ={x:x is a member of A and
x is a member of B}.
 The
intersection of more than
two sets is the set of elements
that belong to each of the sets.
 If A  B = , then we say that A
and B are disjoint.
To help us understand we will use
the Venn Diagram
r2
r1
r3
r4
A B
r2 r3
r4
Inside the 2 circles are shaded.
A B
r2
r3
r4
Which region would be shaded?
Complement
If A is a subset of the universal set U, the
complement of A is the set of elements of
U that are not elements of A. This set is
denoted by A’.
 Using set builder notation,

A'  {x : x U ; x  A}
Venn diagram: Complement
r1
r2
What would be the complement?
Example

Find the complement of each set.
◦ U ={1,2,3,…10} and A ={2,4,6,8}
 A’ = {1,3,5,7,9}
◦ U is the set of cards in a standard
52 card deck, and F is the set of
face cards.
 F’ is the set of nonface cards.
Set difference
 The
difference of sets B and A is
the set of elements that are in B
but not in A. This set is denoted
by B-A.
 Using set builder notation,
◦ B-A = {x:x is a member of B and x
is not a member of A}.
Venn for set difference, B-A
A
r2
B
r3
r4
Which region would be shaded?
Examples of Set difference.
 Find
the set difference.
◦ Find {3, 6, 9, 12}-{x:x is an even
integer}
◦ Therefore we will remove all
the even integers to get {3, 9}