Introduction
Operations on Sets
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Operations on Sets
12-2
Previous Lecture
Sets and elements
Subsets, proper subsets, empty sets
Universe
Cardinality
Power set
Discrete Mathematics – Operations on Sets
12-3
Venn Diagrams
Often it is convenient to visualize various relations between sets.
We use
Venn diagrams
for that.
universe
set
A
B
B
is a subset of
A
Discrete Mathematics – Operations on Sets
12-4
Intersection
The
intersection
of sets
A
and
B,
denoted by
A
∩
B,
is the
set that contains those elements in both
A
and
B.
A
∩
B = { x |
x
∈
A
∧
x
∈
B}
A
B
A
∩
B
Examples
{1,3,5,7}
∩
{2,3,4,5,6} = {3,5}
{Jan.,Feb.,Dec.}
∩
{Jan.,Feb.,Mar.} = {Jan.,Feb.}
{x |
5
y
x=2y}
∩
{x |
5
y
x=3y}
=
{x |
5
y
x=6y}
+
+
=
∩
Z
Q
Z
Discrete Mathematics – Operations on Sets
12-5
Union
The
union
of sets
A
and
B, denoted by
A
∪
B,
is the set that
contains those elements that are in A
or in
B.
A
B
Examples
A
∪
B = { x |
x
∈
A
∨
x
∈
B}
A
∪
B
{Mon,Tue,Wed,Thu,Fri}
∪
{Sat,Sun} = {Mon,Tue,Wed,Thu,Fri,Sat,Sun}
{1,3,5,7}
∪
{2,3,4,5,6} = {1,2,3,4,5,6,7}
Discrete Mathematics – Operations on Sets
12-6
Disjoint Sets and Principle of Inclusion-Exclusion
Sets
A
and
B
are said to be
disjoint
if
A
∩
B =
∅
.
Sets
{Mon,Tue,Wed,Thu,Fri} and
{Sat,Sun}
are disjoint.
Principle of inclusion-exclusion.