Set Operations (Section 1.
5)
An operation in mathematics is a rule for getting an answer. In arithmetic, the four basic
operations are addition, subtraction, multiplication, and division. Notice that these familiar
operations operate on numbers and not on sets.
Set Operations operate on sets. The two basic set operations are Union ∪ and Intersection ∩.
Union of Sets
Union of two sets: The set of all the elements that are in either of the two sets being joined.
Example: For A = {4, 5} and B = {5, 6, 7}, then, A ∪ B = { 4, 5, 6, 7}
Notice that you do not write the 5 in the answer set twice.
Venn Diagram: A B
Notice all of A is shaded
As well as all of B.
Mathematical meaning of “OR”: All of the elements in A or in B or in both.
Intersection of Sets
Intersection of two sets: The set of only the elements that are in common between the two sets.
Intersection is the OVERLAPPED part.
Example: For A = {4, 5} and B = {5, 6, 7}, then, A ∩ B = {5}
Venn Diagram:
A B
Mathematical meaning of “AND”: All of the elements that are in both set A AND set B (at the same time).
MSUM Liberal Studies Course Math 102– Fall 2009. Page 10
Guided Notes to Accompany Text: Mathematics All Around by Pirnot
Complement of a Set
Complement of a set: The complement of any set is the set of all the elements in the given
universal set, U, that are not in the set A. You can think of “the complement of A” as “not A”.
The symbol for complement of a set is the prime ′ . So A′ means “the complement of set A”.
In set builder notation A′ =
{x : x ∈ U but x ∉ A}
U A If set A is represented by the circle and the
Universe is represented by the rectangle,
then A′, (the complement of A or
sometimes called “not A”) is represented by
the shaded region.
Difference of Two Sets
The difference of sets B and A, written B − A , U
is the set of elements that are in that are in B but not in A. A B
Shade B − A in the Venn diagram at the right.
Write B − A using set builder notation:__________________________________
Practice: Use U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Find these sets with respect to this U.
a. A = { 1, 2, 3}. A′ = _______________ b. T = {0, 2, 4, 6, 8} T′ = __________
c. H = ∅. H′ = ______________ d. U′ = __________________________
Let A = {0, 1} B = { 1, 2, 3, 4 } C = {2, 4, 6, 8 . . . }
Draw the Venn diagram for each of the following and then write the resulting set in list form.
a. A∪ B b. A ∩ B c. C − B
MSUM Liberal Studies Course Math 102– Fall 2009. Page 11
Guided Notes to Accompany Text: Mathematics All Around by Pirnot
A Handy Relationship to Notice
Complete the Venn Diagrams for each of the statements below and then compare the results:
( A ∪ B) A′ B′
U U U
A B A B A B
( A ∪ B )′ A′ ∩ B′
U U
A B A B
( A ∪ B )′ compared to A′ ∩ B′ :_______________________________________________
_________________________________________________________________________
_________________________________________________________________________
Now complete this description of the shaded region U
of this Venn diagram: A B
_Everything except ______________________________
_______________________________________________
Write three different set operation expressions that would result in the shaded region above.
i) ii) iii)
MSUM Liberal Studies Course Math 102– Fall 2009. Page 12
Guided Notes to Accompany Text: Mathematics All Around by Pirnot
De Morgan’s Laws for Set Operations
B ∩C A∪ B A∪C
A B A B A B
C C C
A ∪ (B ∩ C) ( A ∪ B) ∩ ( A ∪ C )
U U A B
A B
C C
Comparison:_______________________________________________________________
_________________________________________________________________________
Complete the Venn Diagram for each of the statements below and then compare the results:
A ∩ (B ∪ C) ( A ∩ B) ∪ ( A ∩ C )
A B A B
C C
Comparison:_______________________________________________________________
_________________________________________________________________________
MSUM Liberal Studies Course Math 102– Fall 2009. Page 13
Guided Notes to Accompany Text: Mathematics All Around by Pirnot
