If n(U)= 40, n(A)=17. n(B)=11, and n(A’ n B’)=15, find n(AnB)

Question

If n(U)= 40, n(A)=17. n(B)=11, and n(A’ ∩ B’)=15, find n(A∩B)

Answer 1

|A ∩ B|

 

= |U| - |(A ∩ B)'|  ← Definition of complement.

 

= |U| - |A' ∪ B'| ← DeMorgan's Law

 

= |U| - ( |A'| + |B'| - |A' ∩ B'| ) ← Inclusion-Exclusion Principle.

 

= |U| - (2|U| - |A| - |B| - |A' ∩ B'|) ← Definition of complement ×2.

 

= 40 - (2·40 - 17 - 11 - 15) ← Substitution.

 

= 40 - 37

 

= 3