Find the difference

Question

AnB =0 , and difference between proper subsets of A and proper subsets of B =496

Find the subsets of A

Answer 1

Suppose that A has n elements and B has m elements. The number of proper subsets of A is 2^n-1 and the number of proper subsets of B is 2^m-1. So we have (2^n-1)-(2^m-1)=496 which gives 2^n-2^m=496. We can write this as 2^m(2^(n-m)-1)=496. Note that 2^m is a power of 2, 2^(n-m)-1 is an odd number, and 496=(2^4)(31). So we must have m=4 and 2^(n-m)-1=31 which gives 2^(n-m)=32=2^5, and so n-m=5. Hence n=9 and the number of subsets of A is 2^9.