Operations on numerical sets with intervals

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Given the two sets $A=\{x\mid x\le 2 \ \vee\ 8\le x\le 16\}$ and $B=\{x\mid 2\le x<10\}$, determine:

• (a) $A\cap B$
• (b) $A\cup B$
• (c) $A\setminus B$
• (d) $\bar{A}$ (complement with respect to $\mathbb{R}$)

Solution

Write the sets as unions of intervals: $A=(-\infty,\,2]\cup[8,\,16], \qquad B=[2,\,10).$

(a) $A\cap B$. Intersect $B$ with each piece of $A$: $(-\infty,2]\cap[2,10)=\{2\}$; $\;[8,16]\cap[2,10)=[8,10)$. Hence $A\cap B=\{2\}\cup[8,\,10).$

(b) $A\cup B$. $[2,10)\cup[8,16]=[2,16]$ and $(-\infty,2]\cup[2,16]=(-\infty,16]$: $A\cup B=(-\infty,\,16].$

(c) $A\setminus B$. Remove from $A$ the elements of $B=[2,10)$: $(-\infty,2]\setminus[2,10)=(-\infty,2)$; $\;[8,16]\setminus[2,10)=[10,16]$. Therefore $A\setminus B=(-\infty,\,2)\cup[10,\,16].$

(d) $\bar{A}$. Complement in $\mathbb{R}$ of $(-\infty,2]\cup[8,16]$: $\bar{A}=(2,\,8)\cup(16,\,+\infty).$

$\boxed{A\cap B=\{2\}\cup[8,10)\quad A\cup B=(-\infty,16]\quad A\setminus B=(-\infty,2)\cup[10,16]\quad \bar A=(2,8)\cup(16,+\infty)}$