Question

If $A=\left\{ x:x=3n,n\in Z \right\}$ and\[B=\left\{ x:x=4n,n\in Z \right\}\] then find$A\cap B$.

Answer 1

Hint:First, write all the elements of set A and set B separately using the condition given in the question. Then find all the elements that are common in both the sets. Write these elements in the set of $A\cap B$. Then generalize the final set.

Complete step-by-step answer:
Now, the intersection of two sets $A$ and $B$ is denoted by $A\cap B$, $A\cap B$ is the set that contains all elements of set $A$ that is also in set $B$ or vice versa.
So, to find intersection just find the common terms of both the sets. Disjoint set means that two sets which do not have anything in common than we say $A\cap B$$=\Phi $ (that is empty set or say null set).
So, here we write some elements of both the sets $A=\left\{ x:x=3n,n\in Z \right\}$ and $B=\left\{ x:x=4n,n\in Z \right\}$ .
So,$A=\left\{ ...\text{,-3, 0, 3, 6, 9,}... \right\}$ and $B=\left\{ ...\text{,-4, 0, 4, 8,}... \right\}$
So, we have to find all elements which satisfy the condition.
Let any such element be $a$.
So,$a=3{{n}_{1}}\text{, }{{\text{n}}_{1}}\in Z$ a = 3 and $a=4{{n}_{2}},\text{ }{{\text{n}}_{2}}\in Z$ .
So,$a$ is multiple of $3$ as well as of $4$ respectively. So, elements of $A\cap B$are
$A\cap B$$=\left\{ ...\text{, -24, -12, 0, 12, 24,}.. \right\}$
So, the general term of $A\cap B=C$ is $12n,\text{ }n\in Z$
So, set $A\cap B=C$ in set in set builder form is $A\cap B=C=\left\{ x:x=12n,\text{ n}\in \text{Z} \right\}$ , where $Z$ is integer.

Note: Here, in this question we observe that the intersection of the set $A$ and $B$ is an infinite set as there are infinite multiples of $12$. But also, intersection of two infinite sets can be a finite set and also intersection of two infinite sets can be a null set. But intersection of two finite sets can be either a null set or a finite set.