Section 2.3: Using Venn Diagrams to Study Set Operations
Drawing a Venn Diagram with Two Sets
Region I represents the elements in set 𝐴 that are not in set 𝐵.
Region II represents the elements in set 𝐴 and in set 𝐵. (A
∩
B)
Region III represents the elements in set 𝐵 that are not in set 𝐴.
Region IV represents the elements in the universal set that are in neither
set 𝐴 nor set 𝐵.
Regions I + II + III represents the elements that are in set 𝐴 or in set 𝐵
∪
or in both. (A B)
1
′
EXAMPLE 1 Draw a Venn diagram to illustrate the set (𝐴
∪
𝐵) .
EXAMPLE 1 Draw a Venn diagram to illustrate the set (𝐴
∩
𝐵) .
2
′
EXAMPLE 2 Draw a Venn diagram to illustrate the set 𝐴
3
∪
′
𝐵.
EXAMPLE 2 Draw a Venn diagram to illustrate the set 𝐴
4
∩
′
𝐵.
Drawing a Venn Diagram with Three Sets
Region I represents the elements in set 𝐴 but not in set 𝐵 or set 𝐶.
Region II represents the elements in set 𝐴 and set 𝐵 but not in set 𝐶.
Region III represents the elements in set 𝐵 but not in set 𝐴 or set 𝐶.
Region IV represents the elements in sets 𝐴 and 𝐶 but not in set 𝐵.
Region V represents the elements in sets 𝐴, 𝐵, and 𝐶.
Region VI represents the elements in sets 𝐵 and 𝐶 but not in set 𝐴.
Region VII represents the elements in set 𝐶 but not in set 𝐴 or set 𝐵.
Region VIII represents the elements in the universal set 𝑈 , but not in set
𝐴, 𝐵, or 𝐶.
5
∩
EXAMPLE 3 Draw a Venn diagram to illustrate the set 𝐴 (𝐵
6
∩
′
𝐶) .
Using Venn Diagrams to Decide If Two Sets Are Equal
EXAMPLE 7 Determine if the two sets are equal by using Venn diagrams:
∪
∩
∩
∪
∩
(𝐴 𝐵) 𝐶 and (𝐴 𝐶) (𝐵 𝐶).
7
De Morgans Laws For any two sets A and B,
′
′
′
′
(𝐴
∪
𝐵) = 𝐴
(𝐴
∩
𝐵) = 𝐴
∩
𝐵
∪
𝐵
′
′
EXAMPLE 6 Use Venn diagrams to show that (𝐴
8
∪
′
𝐵) = 𝐴
′
∩
′
𝐵.
EXAMPLE 4 If 𝑈 = {a, b, c, d, e, f, g, h}, 𝐴 = {a, c, e, g}, and 𝐵 =
∪
′
′ ∩
′
{b, c, d, e}, find (𝐴 𝐵) and 𝐴 𝐵
9
EXAMPLE 5 If 𝑈 = {10, 11, 12, 13, 14, 15, 16}, 𝐴 = {10, 11, 12, 13}, and
∩
′
′ ∪
′
𝐵 = {12, 13, 14, 15}, find (𝐴 𝐵) and 𝐴 𝐵 .
10
EXAMPLE 8 In a survey of 100 randomly selected freshmen walking across
campus, it turns out that 42 are taking a math class, 51 are taking an English
class, and 12 are taking both. How many students are taking either a math
class or an English class?
11