# If A, B and C are three finite set then the number of elements in either set A or B or in C is given by

## If A, B and C are three finite set then the number of elements in either set A or B or in C is given by

Definition :- n(A ∪ B ∪ C) = n(A) + n(B) + n(c) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C)
+ n(A C B ∩ c)

Let A,B and C be three sets such that
n(A ∪ B ∪ C) -----(i)
We know that,
n(A ∪ B) = n(A) + n(B) -n(A ∪ B)

Now,by applying the above property in equation(i),we get,
n(A ∪ B ∪ C) = n(A) + [ n(B) + n(c) -n(B ∪ C)] - n[(A ∩ B) ∪ (A ∩ C)]

By applying the distributive property
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C),we get,

n(A) + n(B) + n(C) -n(B ∩ C) -[n(A ∩ B) + n(A ∩ C)] - n(A ∩ B) ∩ n(A ∩ C)
= n(A) + n(B) + n(C) -n(B ∩ C) -[n(A ∩ B) + n(A ∩ C)] - n(A ∩ B ∩C)
= n(A) + n(B) + n(C) -n(B ∩ C) -n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩C)

n(A ∪ B ∪ C) =n(A) + n(B) + n(C) + n(A ∩ B ∩C) -n(B ∩ C) -n(A ∩ B) - n(A ∩ C)

Example 1 :-In a shop, 380 people buy socks, 150 people buy shoes and 200 people buy belt. If there are total 580 people who bought either socks or shoes or belt and only 30 people bought all the three things? So how many people bought exactly two things.

Solution :-

Let S, H and B represent the set of number of people bought socks, shoes and belt respectively. So, n(S) = 380, n(H) = 150, n(B) = 200

This given that,

But this includes the number of people who bought all the three items also. So we have to deduct these number of people from it.
180 – 90 = 90

Hence, 90 people are there who bought exactly two things.

Example 2 :-A = {4, 5, 6}, B = {5, 6, 7, 8} and C = {6, 7, 8, 9} find the value of n(A ∪ B ∪ C)

Solution :-
we know that,n(A ∪ B ∪ C) =n(A) + n(B) + n(C) + n(A ∩ B ∩C) -n(B ∩ C) -n(A ∩ B) - n(A ∩ C)
N(A) = 3
N(B) = 4
N(C) = 4

A n B = {4, 5, 6} n {5, 6, 7, 8} = {5, 6}
n (A n B) = 2

B n C = {5, 6, 7, 8} n {6, 7, 8, 9} = {6, 7, 8}
n (B n C) = 3

C n A = {6, 7, 8, 9} n {4, 5, 6} = {6}
n (C n A) = 1

A n B n C = {6}
n (A n B n C) = 1

= 3 + 4 + 4 – 2 – 3 – 1 + 1
= 12 – 6
= 6