Which of the following is an empty set? This test
Which of the following is an empty set?
- A.
{x | x∈R and x2−1=0}
- B.
{x | x∈R and x2+1=0}
- C.
{x | x∈R and x2−9=0}
- D.
{x | x∈R and x2=x+2}
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Correct answer: B
Solve each set-defining equation over the real numbers:
{x | x∈R and x^2 - 1 = 0}: Factor x^2 - 1 = (x - 1)(x + 1) = 0, so x = 1 or x = -1. The set is {1, -1}.
{x | x∈R and x^2 + 1 = 0}: x^2 = -1 has no real solutions because the square of any real number is ≥ 0. Therefore this set has no real elements and is the empty set.
{x | x∈R and x^2 - 9 = 0}: Factor x^2 - 9 = (x - 3)(x + 3) = 0, so x = 3 or x = -3. The set is {3, -3}.
{x | x∈R and x^2 = x + 2}: Rearrange to x^2 - x - 2 = 0, factor as (x - 2)(x + 1) = 0, so x = 2 or x = -1. The set is {2, -1}.
Conclusion: The only empty set among the given options is {x | x∈R and x^2 + 1 = 0} because x^2 + 1 = 0 has no real solutions.