Two finite sets have m and n elements the total number of subsets of the first is 56 more than the total number of subsets of the second set the values of m and n are?

Answer: Let A has m elements.
Let B has n elements.
Total number of students of A = 2^m
Total number of students of B = 2ⁿ
It is given

2^m – 2ⁿ = 56
2ⁿ (2^m-n – 1) = 56

2ⁿ = even and 2^m-n -1 =0 odd.

Now,

56 = 8×7

=2³ × 2⁷

2ⁿ (2^m-n – 1)

=2³ × 7

⇒ n = 3  

Now,

8(2^m-3 -1) = 8×7

2^m-3 -1 = 7

2^m-3 = 8 = 2³

⇒ m-3 = 3

⇒ m = 6

The value of m and n are 6 and 3 respectively.