If A = xR: |x-2|1, B=xR: x^2-31, C=xR: |x-4|2, and Z is the set of all integers, then th…

Question

If $A = \{x\in\mathbb{R}: |x-2|>1\}$, $B=\{x\in\mathbb{R}: x^2-3>1\}$, $C=\{x\in\mathbb{R}: |x-4|\geq2\}$, and $\mathbb{Z}$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C)^{C}\cap\mathbb{Z}$ is:

Answer: 256

Accepted Answer

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

$B$: $x^2-3>1 \Rightarrow x^2>4 \Rightarrow B = (-\infty,-2)\cup(2,\infty)$.

$C = (-\infty,2]\cup[6,\infty)$.

$$A\cap B\cap C = (-\infty,-2)\cup[6,\infty)$$

Complement: $(A\cap B\cap C)^{C} = [-2,6)$.

Intersect with $\mathbb{Z}$: $\{-2,-1,0,1,2,3,4,5\}$ --- 8 elements.

Number of subsets: $2^8=256$

Solutions · 1