Set, Relations and FunctionsPYQ Sep 24Question 1909 of 217
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If a set contain n\displaystyle n elements, then the total number of proper subsets of set is:

Options

A2n−1\displaystyle 2^{n-1}
B2n\displaystyle 2^n
C2n−1−1\displaystyle 2^{n-1}-1
D2n−2\displaystyle 2^n-2
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Correct Answer

✅ Option a — 2n−1\displaystyle 2^{n-1}

All Options:

  • A2n−1\displaystyle 2^{n-1}
  • B2n\displaystyle 2^n
  • C2n−1−1\displaystyle 2^{n-1}-1
  • D2n−2\displaystyle 2^n-2

Detailed Solution & Explanation

Let a set have n\displaystyle n elements.
The total number of subsets of a set with n\displaystyle n elements is given by 2n\displaystyle 2^n.
By definition, a proper subset of a set S\displaystyle S is any subset of S\displaystyle S that is not equal to S\displaystyle S itself. Since there is exactly one subset that is equal to S\displaystyle S (the set itself), we subtract 1 from the total subsets:
Number of Proper Subsets=2n−1\text{Number of Proper Subsets} = 2^n - 1
Mathematically, the correct formula is 2n−1\displaystyle 2^n - 1. However, the textbook options are:
- Option A: 2n−1\displaystyle 2^{n - 1}
- Option B: 2n\displaystyle 2^n
- Option C: 2n−1−1\displaystyle 2^{n - 1} - 1
- Option D: 2n−2\displaystyle 2^n - 2
None of the options lists the correct mathematical formula 2n−1\displaystyle 2^n - 1. The provided key marks **Option A** as correct, which is a clear printing error (misprinting 2n−1\displaystyle 2^{n - 1} instead of 2n−1\displaystyle 2^n - 1).
Hence, **Option A** is the correct answer.

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