Set, Relations and FunctionsPYQ Nov. 20Question 1890 of 217
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Two finite sets respectively have x\displaystyle x and y\displaystyle y number of elements. The total number of subsets of the first is 56\displaystyle 56 more than the total number of subsets of the second. The value of x\displaystyle x and y\displaystyle y respectively.

Options

A6\displaystyle 6 and 3\displaystyle 3
B4\displaystyle 4 and 2\displaystyle 2
C2\displaystyle 2 and 4\displaystyle 4
D3\displaystyle 3 and 6\displaystyle 6
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Correct Answer

✅ Option a — 6\displaystyle 6 and 3\displaystyle 3

All Options:

  • A6\displaystyle 6 and 3\displaystyle 3
  • B4\displaystyle 4 and 2\displaystyle 2
  • C2\displaystyle 2 and 4\displaystyle 4
  • D3\displaystyle 3 and 6\displaystyle 6

Detailed Solution & Explanation

Let the two finite sets have x\displaystyle x and y\displaystyle y elements respectively.
The number of subsets of the first set is 2x\displaystyle 2^x, and the number of subsets of the second set is 2y\displaystyle 2^y.
According to the problem, the difference between their total subsets is 56\displaystyle 56:
2x−2y=562^x - 2^y = 56
Since 56>0\displaystyle 56 > 0, we must have x>y\displaystyle x > y. We can factor out 2y\displaystyle 2^y from the left side:
2y(2x−y−1)=562^y \left(2^{x - y} - 1\right) = 56
We can express 56\displaystyle 56 as a product of a power of 2 and an odd number:
56=8×7=23×(23−1)56 = 8 \times 7 = 2^3 \times (2^3 - 1)
Equating the components:
2y=23  ⟹  y=32^y = 2^3 \implies y = 3
2x−y−1=23−1  ⟹  x−y=32^{x - y} - 1 = 2^3 - 1 \implies x - y = 3
Substituting y=3\displaystyle y = 3 into x−y=3\displaystyle x - y = 3:
x−3=3  ⟹  x=6x - 3 = 3 \implies x = 6
Therefore, the values of x\displaystyle x and y\displaystyle y are 6\displaystyle 6 and 3\displaystyle 3 respectively.
Let us verify: 26−23=64−8=56\displaystyle 2^6 - 2^3 = 64 - 8 = 56, which is correct.
Hence, **Option A** is the correct answer.

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