If A and B are two finite set then the number of elements in either A or in B is given by

Dec 10, 2021, 16:45 IST

If A and B are two finite set then the number of elements in either A or in B is given by

n(A ∪ B) = n(A) + n(B) -n(A ∩ B)

Definition :-To Understand it easily, lets, say A and B are 2 sets of objects (may be numbers)

A ∪ B contains all the objects contained both in A and B taken one at a time

So, n(AꓴB) implies the number of objects contained in both A and B taken one at a time.
n(A)=No. of objects in A
n(B)=No. of objects in B
A∩B contains objects common between sets A and B

Now, n(A)+n(B) contains total number of objects in A and B including those which are common to A as well as B. Which means, those objects are counted twice as they constitute both A and B.

So, to exclude them, such that we count all the objects in A and B only once, we write…n(A)+n(B)-n(A∩B)
But the above sentence is the definition of n(AꓴB).

Therefore,n(A ∪ B) = n(A) + n(B) -n(A ∩ B)

Example 1 :-In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student liked to play at least one of the two games. How many students like to play both cricket and football?

Solution :-
Let, X be the set of students who like to play cricket
Y be the set of students who love to play football,

n(X) = 24,n(Y) = 16, n(X∪ Y) = 35,n(X∩ Y) = ?
n(X ∪ Y) = n(X) + n(Y) - n(X ∩ Y),
35 = 24 + 16 - n(X ∩ Y)
n(X ∩ Y)= 5

5 students like to play both games.

Example 2 :-In a group of students, 225 students know French, 100 know Spanish and 45 know both. Each student knows wither French or Spanish. How many students are there in the group.

Solution :-
Let F and S denote the no. of students who know French and Spanish, respectively.
Given, n(F) = 225, n(S) = 100, n(F ∩ S) = 45,
Using identity,
n(F ∪ S) = n(F) + n(S) - n(F ∩ S)
=225 + 100 -45
= 325 - 45
= 280