Venn Diagram Formula
Before learning the Venn diagram formula, let us recall what is a Venn diagram. Venn diagrams in math refer to a visual representation of sets. A Venn diagram helps us to visualize the logical relationship between sets and their elements and helps us solve examples based on these sets. In a Venn diagram, intersecting and nonintersecting circles are used to denote the relationship between sets. Let us learn the Venn diagram formula along with a few solved examples.
What is the Venn Diagram Formula?
This is a basic formula from the set theory. For a set A, n(A) represents the number of elements in it. The Venn diagram formula gives the relation between the number of elements of two sets. It is given by:
n(A U B) = n(A) + n(B) – n (A ⋂ B)
Here,
 n(A U B) represents the number of elements present in either one of the sets A or B
 n(A ⋂ B) represents the number of elements present essentially in both sets A and B
Let us see the applications of the Venn diagram formula in the following section.
Solved Examples Using Venn Diagram Formula

Example 1: In a class of 70 students, 45 students like to play Soccer. 52 students like to play Baseball. All the students like to play at least one of the two games. How many students like to play Soccer or Baseball? How many students like to play only Soccer?
Solution: The given information can be shown by Venn diagrams as follows.
Let n (A ⋂ B) = x, n(A) = 45, n(B) = 52,
We know that n(A U B ) = 70
Using the Venn diagram formula,
n(A ⋂ B) = x = n(A) + n(B)  n(A U B)
= 45 + 52  70 = 27
Students who like to play only Soccer = 45  27 =18
Answer: 27 students like to play Soccer or Baseball, and 18 students like to play only Soccer.

Example 2: There are 100 students, 35 like painting and 45 like dancing. 10 likes both. How many of the students like either of them or neither of them?
Solution: Total number of students = 100
Number of students that like painting, n(P) = 35
Number of students that like dancing, n(D) = 45
Number of students who like both, n(P∩D) = 10
Number of students who like either of them using the Venn diagram formula is,
n(PUD) = n(P) + n(D) – n(P∩D)
⇒ 45 + 35  10 = 70
Number of students who like neither = Total students – n(PUD) = 100 – 70 = 30
Answer: 70 students like either of them and 30 students like neither of them.
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