Question

Decide, among the following sets, which sets are subsets of one and another.:
$A=\{x:x\in R$ and $x$ satisfy $x^2-8x+12=0\}$
$B=\{2,4,6\},C=\{2,4,6,8,....\},D=\left\{6\right\}$

Answer 1

Given data:
$A=\{x:x\in R$ and $x$ satisfy $x^2-8x+12=0\}$
$B=\{2,4,6\},C=\{2,4,6,8,....\},D=\left\{6\right\}$
Step 1 : Solve for set A
$\{x:x\in R\text{and}x\text{satisfy}{x}^2-8x+12=0\}$
On solving $x^2-8x+12=0$
$\Rightarrow x^2-6x-2x+12=0$
$\Rightarrow x(x-6)-2(x-6)=0$
$\Rightarrow (x-6)(x-2)=0$
$\Rightarrow x=2,6$
Thus $A=\{2,6\},B=\{2,4,6\},C=\{2,4,6,8,...\},D=\left\{6\right\}$
Since all elements of $A$ are in $B,$
So $A$ is a subset of $B$ i.e., $A\subset B$
All elements of $A$ are in $C\Rightarrow A\subset C$

Step 2: Check for set $B$
$\because$ All elements of set $B$ are in set $C$
$\therefore B\subset C$

Step $3:$ Clearly, $C$ is not a subset of any other given set.

Step $4:$ Check for set $D$
All elements of $D$ are in $A$
All elements of $D$ are in $B$
All elements of $D$ are in $C$
$\therefore D\subset A,D\subset B$ and $D\subset C$

Final answer:
Thus, $A\subset B,A\subset C,B\subset C,D\subset A,D\subset B\text{and}D\subset C$