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Chapter 1: Problem 2

Two finite sets have \(\mathrm{m}\) and \(\mathrm{n}\) element respectively. The total number of subsets of first set is 112 more than the total number of sub sets of the second set. The value of \(\mathrm{m}\) and \(\mathrm{n}\) respectively are (a) 5,2 (b) 4,7 (c) 7,4 (d) 2,5

Short Answer

Expert verified
The correct values for $\mathrm{m}$ and $\mathrm{n}$ are (c) 7, 4 according to the equation \(2^m = 2^n + 112\).

Step by step solution

01

Understand the given information and write the equation we need to solve.

We have two finite sets with m and n elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Using the formula for the number of subsets of a set with k elements : \(2^k\), we can write the equation: \(2^m = 2^n + 112\)
02

Check for possibilities with the given options

Now let's use the equation to figure out which of the options is correct. We need to find the pair of (m, n) that satisfies the above equation: (a) m = 5, n = 2 \(2^5 = 2^2 + 112\) \(32 = 4 + 112\) (Not True) (b) m = 4, n = 7 \(2^4 = 2^7 + 112\) \(16 = 128 + 112\) (Not True) (c) m = 7, n = 4 \(2^7 = 2^4 + 112\) \(128 = 16 + 112\) (True) (d) m = 2, n = 5 \(2^2 = 2^5 + 112\) \(4 = 32 + 112\) (Not True)
03

Write down the correct answer

The only option for (m, n) that satisfies the given equation is option (c) m = 7 and n = 4. Therefore, the correct answer is: (c) 7, 4

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Most popular questions from this chapter

The domain of the function \(\mathrm{f}(\mathrm{x})=\cos ^{-1}\left[\log _{3}(\mathrm{x} / 4)\right]\) is (a) \([4,12]\) (b) \([0,3]\) (c) \([(4 / 3), 4]\) (d) \([(4 / 3), 12]\)

Let \(R\) be a relation on \(\mathrm{N}\) defined by \(\mathrm{R}=\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}+2 \mathrm{y}=8\\}\) The domain of \(\mathrm{R}\) is (a) \(\\{2,4,8\\}\) (b) \(\\{2,4,6,8\\}\) (c) \(\\{2,4,6\\}\) (d) \(\\{1,2,3,4\\}\)

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