Question

Let A and B be two finite set such that there are exactly 144 sets which a subset of A or subset of B. Find the number of elements in AUB.

Answer 1

let the number of elements in A be m and B be n

therefore the total number of subsets of A is 2m and number of subsets of B is 2n

given 2m−2n=960

we know from this equation that m>n

therefore taking n common we get

2n(2m−n−1)=960

as 2(m−n)−1 is odd the even part is only 2n

960 can be written as 26×15

therefore from above equation we can observe that n=5

and $$2^(m-5)-1$=15$

⇒2(m−5)=16 so m=9

therefore n(A)-n(B)=m-n=4

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