A collection of objects is called a Set. Sets formulas form a foundation for many fields of mathematics. In the areas related to statistics, and particularly in probability. Venn diagrams are popularly used to visualize problems in set theory to arrive at a solution by represented them by overlapping circles. Let's learn about sets formulas with a few solved examples in the end.
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These are the basic set of formulas from the set theory. For two sets A and B, sets formula can be given as,
Formula 1: Union formula in set theory indicates the number of elements present in either one of the sets A or B. It is represented as,
n(A U B)
Formula 2: Intersection formula in set theory indicates the number of elements present essentially in both sets A & B. It is represented as,
n(A ⋂ B)
Formula 3: Another important formula relating union and intersection is represented as,
n(A U B) = n(A) + (n(B) – n (A ⋂ B)
Let us have a look at a few solved examples to understand the sets formula better.
Solved Examples Using Sets 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. Using sets formula find how many students like to play Soccer or Baseball ? How many students like to play only Soccer?
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 sets 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: 18 students like to play only soccer.
Example 2: There are 100 students, 35 like painting and 45 like dancing. 10 like both. How many of the students like either of them or neither of them?
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
Using sets formula, number of students who like either of them,
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.