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Chapter 1: Problem 20

If \(\mathrm{U}=\\{1,2,3\\}\) and \(\mathrm{A}=\\{1,2\\}\) then \([\mathrm{P}(\mathrm{A})]^{\prime}=\ldots \ldots \ldots\) (a) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2\\}, \Phi\\}\) (b) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2,3\\}\\}\) (c) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2,3\\}, \Phi\\}\) (d) \(\\{\\{3\\},\\{2,3\\},\\{1,3\\},\\{1,2\\}\\}\)

Short Answer

Expert verified
(b) \(\{\{3\},\{2,3\},\{1,3\},\{1,2,3\}\}\)

Step by step solution

01

Finding the Power Set of A

First, we need to find the power set of A. The power set of A (\(\mathrm{P}(\mathrm{A})\)) includes all possible subsets of A, in this case: \(\Phi\), \(\{1\}\), \(\{2\}\), and \(\{1,2\}\). So, \(\mathrm{P}(\mathrm{A})\) = \(\{\Phi, \{1\}, \{2\}, \{1,2\}\}\).
02

Finding the Power Set of U

Next, we need to find the power set of the universe U (denoted as \(\mathrm{P}(\mathrm{U})\)). U = \(\{1, 2, 3\}\), so the possible subsets are: \(\Phi\), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,2\}\), \(\{1,3\}\), \(\{2,3\}\), and \(\{1,2,3\}\). Thus, \(\mathrm{P}(\mathrm{U})\) = \(\{\Phi, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}\).
03

Finding the Complement of Power Set of A

To find the complement of the power set of A, denoted as \([\mathrm{P}(\mathrm{A})]^{\prime}\), we will take the elements of \(\mathrm{P}(\mathrm{U})\) which are not in \(\mathrm{P}(\mathrm{A})\). So, \([\mathrm{P}(\mathrm{A})]^{\prime}\) = \(\{\{3\}, \{1,3\}, \{2,3\},\{1,2,3\}\}\). Comparing our result with the given options, we find the correct answer is (b) \(\{\{3\},\{2,3\},\{1,3\},\{1,2,3\}\}\).

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Most popular questions from this chapter

\(\mathrm{A}=[-1,1], \mathrm{B}=[0,1], \mathrm{C}=[-1,0]\) \(\mathrm{S}_{1}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{A}, \mathrm{y} \in \mathrm{A}\right\\}\) \(\mathrm{S}_{2}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{A}, \mathrm{y} \in \mathrm{B}\right\\}\) \(S_{3}=\left\\{(x, y) / x^{2}+y^{2}=1, x \in A, y \in C\right\\}\) \(\mathrm{S}_{4}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{B}, \mathrm{y} \in \mathrm{C}\right\\}\) then (a) \(\mathrm{S}_{1}\) is not a graph of a function (b) \(\mathrm{S}_{2}\) is not a graph of a function (c) \(S_{3}\) is not a graph of a function (d) \(\mathrm{S}_{4}\) is not a graph of a function

If \(\mathrm{A}=\\{1,2,3\\}\), then the number of equivalence relation containing \((1,2)\) is (a) 1 (b) 2 (c) 3 (d) 8

The domain of the function \(\mathrm{f}(\mathrm{x})=[1 / \sqrt{(|\mathrm{x}|-\mathrm{x})]}\) is (a) \((-\infty, \infty)\) (b) \((0, \infty)\) (c) \((-\infty, 0)\) (d) \((-\infty, \infty)-\\{0\\}\)

If A is the set of even natural numbers less than 8 and \(\mathrm{B}\) is the set of prime numbers less than 7, then the number of relations from \(\mathrm{A}\) to \(\mathrm{B}\) is (a) \(2^{9}\) (b) \(9^{2}\) (c) \(3^{2}\) (d) \(2^{9}-1\)

Let \(\mathrm{R}\) be a relation in \(\mathrm{N}\) defined by \(\mathrm{R}=\left\\{\left(1+\mathrm{x}, 1+\mathrm{x}^{2}\right) / \mathrm{x} \leq 5, \mathrm{x} \in \mathrm{N}\right\\}\) which of the following is false? (a) \(\mathrm{R}=\\{(2,2),(3,5),(4,10),(5,17),(6,25)\\}\) (b) Domain of \(\mathrm{R}=\\{2,3,4,5,6\\}\) (c) Range of \(\mathrm{R}=\\{2,5,10,17,26\\}\) (d) (b) and (c) are true
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