Question

Give a reason for the following statements:-
(i)$\left\{2,3,5,7,9,11\right\}\ne \left\{x:xisaprimenumber,x<12\right\}$
(ii)$\left\{1,64,125\right\}\ne \left\{x:xisaperfectsquareandperfectcube,x\le 125\right\}$

Answer 1

$\left(i\right)$ Let $A=\{2,3,5,7,9,11\}$

Consider the set $B=\{x:xisaprimenumberx<12\}$

$=\{2,3,5,7,11\}$

Here $A$ is not a subset of $B$ and $B$ is a subset of $A$

Thus,$\{2,3,5,7,9,11\}\ne \{2,3,5,7,11\}$

Hence $\{2,3,5,7,9,11\}\ne \{x:xisaprimenumberx<12\}$

$\left(ii\right)\{1,64,125\}$ is a set of cubic numbers from $1,4,5$

Consider the set $\{x:xisaperfectsquareandperfectcube,x\le 125\}$

$=\{1,64\}$

$\{1,64\}$ is a subset of $\{1,64,125\}$

But $\{1,64,125\}$ is not a subset of $\{1,64\}$

$\therefore \{1,64,125\}\ne \{1,64\}$