Given n(A) = 285, n(B) = 195, n(∪) = 500, n(A ∪ B) = 410, find n(A’ ∩ B’).

Here we are provided with the cardinality of two sets A and B with and a universal set U and the cardinality of union of both the sets A and B is also provided.

The cardinality of the sets are as follows,

n(U) = 500

n(A) = 285

n(B) = 195

n(A ⋃ B) = 410

From De Morgan’s Law we know that if U is an universal set and it contain two set named A and B then,

n(A’ ⋂ B’) = n(A ⋃ B)’......(i)

To find the value of n(A' ∩ B') we first we have to find the value of (A ⋃ B)'.

We know that when the cardinality of union of two sets is given we can find the cardinality of their union difference by using the formula given below, The cardinality of (A ⋃ B)' is given by

n(A ⋃ B)’ = n(U) – n(A ⋃ B)

So using the formula and data given

n(A ⋃ B)’ = n(U) – n(A ⋃ B)

= 510 – 410

= 90

Now from De Morgan’s law,

n(A’ ⋂ B’) = n(A ⋃ B)’

= 90