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Chapter 1: Problem 4

If \(\mathrm{X} \cup\\{3,4\\}=\\{1,2,3,4,5,6\\}\) the which of the following is true (a) Smallest set \(\mathrm{X}=\\{1,2,5,6\\}\) (b) Smallest set \(\mathrm{X}=\\{1,2,3,5,6\\}\) (c) Smallest set \(\mathrm{X}=\\{1,2,3,4\\}\) (d) Greatest set \(\mathrm{X}=\\{1,2,3,4\\}\)

Short Answer

Expert verified
The correct answer is (a) smallest set X = {1,2,5,6}.

Step by step solution

01

Identify the Given Information

We know that the union of set X and set {3,4} results in the set {1,2,3,4,5,6}.
02

Recall the Union Property

The union of two sets A and B, denoted as A ∪ B, is a new set formed by combining all the elements of sets A and B (without duplication).
03

Determine Set X

Since we know that X ∪ {3,4} = {1,2,3,4,5,6}, we can work backwards to determine set X by subtracting {3,4} from the resulting set. First, we include {3,4} in set X. We are left with {1,2,5,6} which would be the smallest possible set X.
04

Evaluate the Options

Let's compare our result with the given options: (a) Smallest set X = {1,2,5,6} (b) Smallest set X = {1,2,3,5,6} (c) Smallest set X = {1,2,3,4} (d) Greatest set X = {1,2,3,4}
05

Choose the Correct Option

Since our result matches option (a), we conclude that the smallest set X = {1,2,5,6}.

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Most popular questions from this chapter

If \([\mathrm{x}]\) stands for the greatest integer function then the value of \([(1 / 5)+(1 / 1000)]+[(1 / 5)+(2 / 1000)]+\ldots \ldots+[(1 / 5)\) \(+(999 / 1000)]\) is (a) 199 (b) 201 (c) 202 (d) 200

The inverse of \(\left[\left(7^{x}-7^{-x}\right) /\left(7^{x}+7^{-x}\right)\right]\) is (a) \((1 / 2) \log _{7}[(1+\mathrm{x}) /(1-\mathrm{x})]\) (b) \(\log _{7}[(1-x) /(1+x)]\) (c) \(\log _{(1 / 2)}[(1-x) /(1+x)]\) (d) \((1 / 2) \log _{e}[(1+x) /(1-x)]\)

Let \(\mathrm{A}=\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=\mathrm{e}^{\mathrm{x}}, \mathrm{x} \in \mathrm{R}\right\\}, \mathrm{B}=\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=\mathrm{e}^{-\mathrm{x}}, \mathrm{x} \in \mathrm{R}\right\\}\) then (a) \(\mathrm{A} \cap \mathrm{B}=\Phi\) (b) \(\mathrm{A} \cap \mathrm{B} \neq \Phi\) (c) \(A \cup B=R\) (d) \(A \cup B=A\)

There are three-three sets are given in column-A and column-B \begin{tabular}{l|l} \(\frac{\text { Column-A }}{(1)\\{\mathrm{L}, \mathrm{A}, \mathrm{T}\\}}\) & (A) \(\left\\{\mathrm{x} / \mathrm{x} \in \mathrm{z}, \mathrm{x}^{2}<5\right\\}\) \\\ (2) \(\left\\{\mathrm{x} \in \mathrm{z} / \mathrm{x}^{3}-\mathrm{x}=0\right\\}\) & (B) \(\\{\mathrm{x} / \mathrm{x}\) is a letter of the word LATA \(\\}\) \\ (3) \(\\{-2,-1,0,1,2\\}\) & (C) \(\\{\sin 0, \sin (3 \pi / 2), \tan (5 \pi / 4)\\}\) \end{tabular} Which one of the following matches is correct? (a) \(1-\mathrm{A}, 2-\mathrm{B}, 3-\mathrm{C}\) (b) \(1-\mathrm{B}, 2-\mathrm{A}, 3-\mathrm{C}\) (c) \(1-\mathrm{B}, 2-\mathrm{C}, 3-\mathrm{A}\) (d) \(1-\mathrm{A}, 2-\mathrm{C}, 3-\mathrm{B}\)

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) \((-\infty, \infty)-\\{0\\}\)
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