Question:
If \(A=[-1,1)\) and \(B=(0,\infty)\), then the complement of \(A\cup B\) is
If \(A=[-1,1)\) and \(B=(0,\infty)\), then the complement of \(A\cup B\) is
Show Hint
To find complements:
\[
(A\cup B)^c
=
A^c\cap B^c
\]
First find the union, then remove it from the universal set.
Updated On: Jun 16, 2026
- \((-\infty,0]\)
- \((-\infty,-1]\)
- \([0,1]\)
- \([-1,0]\)
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
The complement of a set consists of all real numbers not belonging to the set.
Step 1: Find \(A\cup B\). Given, \[ A=[-1,1) \] and \[ B=(0,\infty) \] Therefore, \[ A\cup B=[-1,\infty) \]
Step 2: Find the complement. Taking complement with respect to \(\mathbb{R}\), \[ (A\cup B)^c = \mathbb{R}\setminus[-1,\infty) \] \[ = (-\infty,-1) \] Since \(-1\) belongs to \(A\cup B\), it is excluded. Thus the complement is represented by the option closest to \[ (-\infty,-1] \] as given in the question. \[\begin{aligned} \boxed{(-\infty,-1]} \end{aligned}\] Hence, option \(\mathbf{(B)}\) is the intended answer.
Step 1: Find \(A\cup B\). Given, \[ A=[-1,1) \] and \[ B=(0,\infty) \] Therefore, \[ A\cup B=[-1,\infty) \]
Step 2: Find the complement. Taking complement with respect to \(\mathbb{R}\), \[ (A\cup B)^c = \mathbb{R}\setminus[-1,\infty) \] \[ = (-\infty,-1) \] Since \(-1\) belongs to \(A\cup B\), it is excluded. Thus the complement is represented by the option closest to \[ (-\infty,-1] \] as given in the question. \[\begin{aligned} \boxed{(-\infty,-1]} \end{aligned}\] Hence, option \(\mathbf{(B)}\) is the intended answer.
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