Question

If X and Y are two non -empty sets then what is (X-Y) ' is equal to what?

:::-))) prove it by using properties of sets not with example​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

X and Y are two non - empty sets.

We know,

From properties of sets,

$\rm :\longmapsto\:X - Y$

$\rm \:  =  \:  \: X \: \cap \: Y'$

and

By De Morgan's Law,

$\rm :\longmapsto\: {(X \: \cap \: Y)}^{'} = X' \: \cup \: Y'$

Also,

$\rm :\longmapsto\: {( {X}^{'} )}^{'} = X$

Now, Consider

$\rm :\longmapsto \: {(X - Y)}^{'}$

$\rm \:  =  \:  \: {(X \: \cap \: {Y}^{'})}^{'}$

$\rm \:  =  \:  \: {X}^{'} \: \cup \: {( {Y}^{'} )}^{'}$

$\rm \:  =  \:  \: {X}^{'} \: \cup \: Y$

So,

$\rm :\longmapsto \: {(X - Y)}^{'} = {X}^{'} \: \cup \: Y$

So, option (c) is correct.