Question

If n(A) = 7, n(B) = 6, n(A U B) = 10, then find n(
AB).
​

Answer 1

Ifn(A)=7,n(A∪B)=11,andn(B)=5,thenwhatisn(A∩B)?

Solution

n(A)=7

n(B)=5

n(A∪B)=11

(A∩B)=?

Let the intersection of A and B = n,n(Aonly)=7−n,n(Bonly)=5−n

n(AUB)=n(Aonly)+n(Bonly)+n(AnB)

After you replace the equations in the formula above,

n(AUB)=n(Aonly)+n(Bonly)+n(AnB)

n(AUB)=7−n+5−n+n

Remember our  n(AUB)=11,,

11=7−n+5−n+n

Revise the formula for easy understanding starting from  n,so

n+5−n+7−n=11

n−n−n+5+7=11

The first two n (s) cancel out and we remain with one, as shown below

5+7−n=11

But  5+7=12

Therefore,

12−n=11

Here n crosses to become a positive (+n) and 11 crosses to the opposite side to become a negative (-11) as shown below,

12−n=11

12−11=n

but  12−11=1

1=n

n=1

But remember we said let the intersection of A and B = n

so n(AnB)=n

Theren(AnB)=1.

So  Ifn(A)=7,n(A∪B)=11,andn(B)=5,thenn(A∩B)=1.