Venn diagrams are visual representations of sets. A rectangle represents the Universal set, U. Each set is represented by a circle or ellipse inside this rectangle. The circles can show the relationships among sets.
Venn diagram 1 is a three-circle Venn diagram.
- The green circle is set A, the blue circle is set C and the red circle is set B.
- The region outside a circle shows the complement of that set.
- Two circles together represent the union of the two sets.
- The overlapping part of two circles represents the intersection of the two sets.
The diagram allows us to depict A ⋂ B, B ⋂ C, C ⋂ A, and even A ⋂ B ⋂ C !
Take a look at Venn diagram 2. Graeter’s polled its customers on their favorite flavor of ice cream: chocolate or vanilla. 205 customers said they liked chocolate, 100 customers said they liked vanilla, while 35 customers said they liked both chocolate and vanilla. How many customers said they liked only chocolate?
Let C represent the set of all customers who like Chocolate and V represent those who like vanilla. The problem states that
|C| = 205, |V| = 100, |C ⋂ V| = 35
We need to find the number of customers who liked only chocolate, or |C - V|. This is represented in the diagram as:
|C| = 205 = x + 35 or x = 170.
Answer: 170 customers like only chocolate.
How many customers only like vanilla?
|V| = 100 = y + 35, or y = 65
Answer: 65 customers liked only vanilla.
From Venn diagram 2, you can see that for sets C, V:
|C ∪ V| = |C| + |V| - |C ⋂ V|
This is an important property of cardinality!
Create an expression that represents the colored portion of Venn diagram 3.
The shaded portion represents that part of set B that is not in set A. It can also be represented as the intersection of B with the complement of A.
Answer: The shaded portion is (B - A) or (Ac ⋂ B)