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Set Sizes When One Has 56 More Subsets Than the Other | JEE Advanced Sets

JEE Maths question with a full step-by-step solution.

Question

Let the finite sets

A

and

B

have

'm'

and

'n'

elements respectively. Given that the total number of subsets of

A

56

more than the total number of subsets of

B

, find the value of

(m+n)

13

9

correct

10

15

Solution

Step 1: A finite set with

k

elements has

2^{k}

subsets, so

2^{m} = 2^{n} + 56 \ \Longrightarrow\ 2^{m}-2^{n} = 56

Step 2:

2^{m}-2^{n} = 56 > 0

, so

m > n

and

2^{n}\left(2^{m-n}-1\right) = 56

56 = 8 \times 7 = 2^{3}\times 7

Step 3:

m-n \geq 1

, so

2^{m-n}-1

is odd and must take the whole odd part, leaving the whole power of two to

2^{n}

2^{n} = 2^{3} \ \Longrightarrow\ n = 3, \qquad 2^{m-n}-1 = 7 \ \Longrightarrow\ 2^{m-3} = 8

This split is the only one possible, so

m

and

n

are unique. Step 4:

2^{m-3} = 2^{3} \ \Longrightarrow\ m-3 = 3 \ \Longrightarrow\ m = 6

m+n = 6+3 = 9

Answer: (2).

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