Universal Set
Section 2.3
 A universal set for a particular problem is a set which contains all the
elements of all the sets in the problem.
 A universal set is often denoted by a capital U.
Venn diagrams – is a pictorial representation of set(s).
 Venn diagrams are generally created by drawing a rectangle to represent
the Universal set.
 Then between 1 and 3 overlapping circles are drawn inside the rectangle to
represent the sets contained within the universal set.
 3 circles are generally the maximum number of circles inside universal set.
 The elements of the sets of the universe are placed in the universal set
starting with the overlapping areas.
Here is a Venn diagram that represents sets U, A and B.
 Write down the elements of each set
U={
}
A={
}
B={
}
Here is a Venn diagram that represents the sets U, S and E.
 Write down the elements of each set
U={
}
S={
}
E={
}
Here is a Venn diagram that represents the sets U, A, B and C.
 Write down the elements of each set
U={
}
A={
}
B={
}
C={
}
Definition of Set Complement
The complement of set A, symbolized by A′, is the set of all the elements in the
universal set that are not in the set A.
A′ is read “A complement,” or “A prime.”
 Find A’ for the set A depicted in the following Venn diagram.
Find the set E’ for the set E depicted in the following Venn diagram.
Find the set C’ for the set C depicted in the following Venn diagram.
Problems 15 -18 from the text want us to draw Venn diagrams for more
complex sets.
Let’s try #16, construct a Venn diagram for the following sets
U = {microwave, washing machine, dryer, refrigerator, dishwasher, cd player,
vcr, camcorder, tv, computer}
A = {microwave, washing machine, dishwasher, computer, tv}
B = {washing machine, dryer, refrigerator, cd player, computer, tv}
Definition of Set Intersection
The intersection of sets A and B, symbolized by A∩B, is the set containing all the
elements that are common to both A and B.
A∩B is read “A intersect B,” or “A AND B.”
Find 𝐴 ∩ 𝐵 from the following Venn diagram.
Find 𝑆 ∩ 𝐸 from the following Venn diagram
Find 𝐵 ∩ 𝐶 from the following Venn diagram
Let’s try some more complicated problems
Find for A and B depicted above.
𝐴 ∩ 𝐵′
𝐴′ ∩ 𝐵′
(𝐴 ∩ 𝐵)′
Find the following for sets S and E depicted above.
𝑆 ∩ 𝐸′
𝐸′ ∩ 𝑆′
(𝐸 ∩ 𝑆)′
Find
𝐴 ∩ 𝐵′
𝐴∩𝐵∩𝐶
(𝐴 ∩ 𝐵)′
Example: Let U = {1,2,3,4,5,6,7,8,9} and A = {1,3,5,7,9} and B = {2,3,4,5}
Create a Venn diagram to model the sets.
Find
𝐴 ∩ 𝐵′
𝐴′ ∩ 𝐵′
(𝐴 ∩ 𝐵)′
Let’s create a new U and new sets A, B and C.
U = {1,2,3,4,5} A = {1,2,3} B = {3,4,5} C = {2,3}
Create a Venn diagram to model the sets.
Find the following
 A’
 B’
 A’∩C
 A∩B’∩C
 (A∩B)’
Definition of Set Union
The union of set A and set B, symbolized by A∪B, is the set containing all the
elements that are members of set A or of set B (or of both sets).
A∪B is read “A union B,” or “A OR B.”
For the Venn diagram above find
 𝐴∪𝐵
 𝐴 ∪ 𝐵′
For the Venn diagram above find
 𝐴∪𝐶
 𝐴∪𝐵∪𝐶
 𝐵 ∪ 𝐶′
Let U = {1,2,3,4,5,6,7,8,9} and A = {1,3,5,7,9} and B = {2,3,4,5}
Draw a Venn diagram and find the following:
 A∪B
 A’ ∪ B
 (A ∪ B)’
Let’s use these sets again: U = {1,2,3,4,5} A = {1,2,3} B = {3,4,5} C = {2,3}
Draw a Venn diagram and find the following:
 A∪B∩𝐶
 (𝐴 ∩ 𝐵)′ ∪ 𝐶
 𝐴 ∪ (𝐶 ∩ 𝐵)′
Union Rule for Sets (Not in the section, but I may need to cover it later)
The relationship between sets A, B, A∪B, and A∩B is given by the union rule:
n(A∪B) = n(A) + n(B) – n(A∩B)
Why? (I will show you how to verify this using a Venn Diagram.)
Two Possible Union Rule Test Problems:
1. If n(A) = 5, n(B) = 8, and n(A∩B) = 2, find n(A∪B).
Answer:
2. If n(A)=12, n(A∪B)=22, and n(A∩B)=10, find n(B).
Answer: