202201260719 Problem 1.57

Let $A=\{ x:x^2-3x+2=0\}$ , $B=\{ x:x^2\le 16\}$ . Determine whether or not $A\subset B$ .

_{Extracted from M. R. Spiegel. (1969). Schaum’s Outline of Theory and Problems of Real Variables}

Set $A$ and set $B$ are described according to property method; if described by the roster method:

$\begin{aligned} A & =\{ 1,2\} \\ B & = \{ -4,-3,-2,-1,0,1,2,3,4\} \\ \end{aligned}$

Obviously set $A$ is a subset of set $B$ .

Roughwork.

(set $A$ )

$\begin{aligned} x^2-3x+2 & = 0 \\ (x-2)(x-1) & = 0 \\ x & = 1,2 \\ \end{aligned}$

(set $B$ )

$\begin{aligned} x^2 & \le 16 \\ x^2 - 16 & \le 0 \\ (x-4)(x+4) & \le 0 \\ -4\le x & \le 4 \\ \end{aligned}$