Question

Decide among the following sets , which are subsets of which :

A = {x : x satisfies $x^2-8x+12=0$},

B = {2, 4, 6}, C = {2, 4, 6, 8, .....}, D = {6}.

Answer 1

We have,

A = {x : x satisfies $x^2-8x+12=0$}

= {x : $x^2-6x-2x+12=0$}

= {x : x (x-6)- 2(x-6)=0}

= {x : (x-6)(x-2)=0}

={x :x = 6, 2}

= {6, 2}

B = {2, 4, 6}

C = {2, 4, 6, 8, ....}

D = {6}

We know that if E and F are two sets, then E is a subset of F, i.e., E $\subseteq$ F if x $ϵ$ E $\Rightarrow$ x ~\epsilon~F. E is called a proper subset of F if E is strictly contained in F and is denoted by E $\subset$ F.

Clearly,

D $\subset$ A { $\because$ 6 $ϵ$ A}

A $\subset$ B {$\because$ 2, 6 $ϵ$ A and they also belong to B}

Similarly, B $\subset$ C

Hence, D $\subset$ A $\subset$ B $\subset$ C.