Which of the following sets is/are empty?

{x : x \in R, x^{2} - 4 = 0}

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{x : x \in R, x^{4} + 4 = 0}

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{x : x \in R, x^{3} = 1}

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{x : x \in R, x^{8} + x^{4} + 1 = 0}

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Solution

The correct options are
B ${x : x \in R, x^{4} + 4 = 0}$
D ${x : x \in R, x^{8} + x^{4} + 1 = 0}$
A) $x = 2, - 2$ is a solution
B) $x^{4}$ is positive always and 4 is also positive.The sum of two positive numbers is always positive.Hence x doesn't have a solution.
C) $x^{3}$ =1 $\Rightarrow x = 1$ .
D) $x^{8} + x^{4} + 1$ is always positive because the individual terms are positive.Hence x here doesn't have a solution.

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