If A and B are two sets and U is the universal set such that n(∪) = 700, n(A) = 200, n(B) = 300 and n (A ∩ B) = 100,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 intersection of both the sets A and B is also provided.

The cardinality of the sets are as follows,

n(U) = 700

n(A) = 200

n(B) = 300

n(A ∩ B) = 100

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 U B)'.

We know that when the cardinality of two sets and the cardinality of their intersection is given we can find the cardinality of their union using the formula given below,

n(A ⋃ B) = n(A) + n(B) –n(A ⋃ B)

When we have the cardinality of union of two set the cardinality of their union difference (A U B)' can be found using the formula below

n(A ⋃ B)’ = n(U) – n(A ⋃ B)

*(A U B)' show the elements which are only the part of the universal set U and doesn’t exist in (A U B).

So using formula and putting values,

n(A ⋃ B) = n(A) + n(B) –n(A ⋃ B)

= 200 + 300 – 100

= 500 – 100

= 400

and n(A ⋃ B)’ = n(U) – n(A ⋃ B)

= 700 – 400

= 300

So putting these values in eqⁿ (i) we will find

n(A’ ⋂ B’) = n(A ⋃ B)’

n(A' ∩ B') = 300