Set Operations and Venn
Diagrams
2.2 – 2.3
Intersection of Sets
The intersection of sets A and B, denoted by A  B ,
is the set of all elements that are common to both.
That is, A  B  {x | x  A and x  B} .
Union of Sets
The union of sets A and B, denoted by A  B , is the
set of all elements that are either in A or B or in both.
That is, A  B  {x | x  A or x  B} .
Let A = {1,2,3,4,5}, B = {1,3,4,6}, and C = {1,6,7}. Find the
following:
1 c. B  C = {1, 6}
2 a. A  B = {1, 2, 3, 4, 5, 6}
3 b. A  ( B  C ) = {1, 2, 3, 4, 5, 6}
A
2
5
B
3
4
1
6
7
C
U
Complement of a Set
Let Ų be the universal set, and let A be a subset of U .
The complement of A, denoted by A’, is the set of
elements in U that are not in A.
That is, A'  {x | x  U and x  A}.
This set is also symbolized by U – A .
Let U = {a,b,c,d,e,f}, A = {a,c,e}, B = {b,d,e,f}, and
C = {a,b,d,f}. Find each specified set.
16 b. ( A  B)' = {a, b, c, d, f}
17 a. ( A  B)' = Ø
18 a. ( A  B)  C' = {c, e}
19 b. C  ( A  B)' = {a, b, c, d, f}
Difference of Sets
If A and B are two sets, the difference of A and B,
denoted by A – B, is the set of all elements that are in
A and not in B.
That is, A - B  {x | x  A and x  B}.
Let U = {1,2,3,4,5}, A = {2,3,4}, and B = {1,4,5}. Find each
specified set.
27 b. U  B = {2, 3}
28 b.
B  A = {1, 5}
A
B
5
2
4
3
1
U
Use the numbered regions of the diagram below to identify
each specified set.
a.
A  B = {1,2, 3,5,6,7}
b.
B  C = {6,7}
c.
A  B  C = {7}
d.
B ' = {1,4,5,8}
e.
A  C   B
= {1,4,5}