Question

List the elements of $A\cap B$, if $\mathrm{A}=\{\mathrm{x}\mid \mathrm{x}+3=0,\mathrm{x}\in \mathrm{\mathbb{N}}\},\mathrm{B}=\{\mathrm{x}\mid \mathrm{x}\le 3,\mathrm{x}\in \mathrm{W}\}$.

Answer 1

Find the intersection of given sets:

Given, the two set $\mathrm{A}=\{\mathrm{x}\mid \mathrm{x}+3=0,\mathrm{x}\in \mathrm{\mathbb{N}}\},\mathrm{B}=\{\mathrm{x}\mid \mathrm{x}\le 3,\mathrm{x}\in \mathrm{W}\}$

We know that, intersection of sets is the set of elements which are common in all the given sets.

Now, in set $A$ we have such natural number $x$ for which the condition $x+3=0$ holds,

$$\begin{gathered} \because \mathrm{x}+3=0\mathrm{for}\mathrm{some}\mathrm{x}\in \mathrm{\mathbb{N}} \\ \Rightarrow \mathrm{x}=-3\mathrm{for}\mathrm{some}\mathrm{x}\in \mathrm{\mathbb{N}} \end{gathered}$$

Which is not possible for any natural number that means set $A$ does not have any element.

$\therefore \mathrm{A}=\mathrm{\varphi }$

We know that, intersection of any set with an empty set will be an empty set.

Hence, $A\cap B=$$\mathrm{\varphi }$.