Let A = xR: |x+1|

Question

Let $A = \{x\in\mathbb{R}: |x+1|<2\}$ and $B = \{x\in\mathbb{R}: |x-1|\geq2\}$. Then which one of the following statements is NOT true?

Accepted Answer

$A=(-3,1)$, $B=(-\infty,-1]\cup[3,\infty)$.

- $A-B = (-1,1)$  (matches A)
- $A\cap B = (-3,-1]$  (matches C)
- $A\cup B = (-\infty,1)\cup[3,\infty) = \mathbb{R}-[1,3)$  (matches D)
- $B-A$: removing $A$ from $(-\infty,-1]$ leaves $(-\infty,-3]$ (since $(-3,-1]\subset A$); removing $A$ from $[3,\infty)$ leaves it unchanged. So $B-A = (-\infty,-3]\cup[3,\infty) = \mathbb{R}-(-3,3)$, not $\mathbb{R}-(-3,1)$ as option B claims.

So option B is the false statement.

Solutions · 1