If $A=\left\{ a\text{, b, c, d, e, f} \right\}$ , $B=\left\{ \text{c, e, g, h} \right\}$ and $C=\left\{ a\text{, e, m, n} \right\}$, find:
(iv) $B\cap C$
(v) $C\cap A$
(vi) $A\cap B$

Hint: To find the intersection of two given sets$A$ and set$B$ that is $A\cap B$ we find all elements which are in set $A$ as well as in set $B$.That is we have to find the common elements.

Complete step-by-step answer:
Now, as given in the question,

$A=\left\{ a\text{, b, c, d, e, f} \right\}$

$B=\left\{ \text{c, e, g, h} \right\}$

$C=\left\{ a\text{, e, m, n} \right\}$

 So, (iv) $B\cap C$

(vi) For finding $B\cap C$ we have to find elements in set $B$ which are also in set $C$ that are common to both sets.

Now, elements in set $B=\left\{ \text{c, e, g, h} \right\}$ and elements in set $C=\left\{ \text{a, e, m, n} \right\}$

So, common elements in set $B$ and set $C$ are : $c,e$

So, we can say $B\cap C=\left\{ e \right\}$

Therefore, $B\cap C=\left\{ e \right\}$

(v) $C\cap A$

For finding $C\cap A$, we follow the same process as in part (iv) that is we find elements which are common in both the set $A$ and set $C$ .

Now, elements in set $C=\left\{ \text{a, e, m, n} \right\}$ and elements in set $A=\left\{ \text{a, b, c, d, e, f} \right\}$ .

So, common elements in set $A$ and set $C$ are : $a,e$

So, we can say $C\cap A=\left\{ a,e \right\}$

Therefore, $C\cap A=\left\{ a,e \right\}$

(vi) $A\cap B$

Again, in this case we have to find $A\cap B$.For this we follow the procedure as earlier in part (v). We have to find all elements in set $A$ which are also in set $B$ , that is common elements in both set $A$ and set $B$.

Now , elements in set $A=\left\{ a\text{, b, c, d, e, f} \right\}$and elements in set $B=\left\{ \text{c, e, g, h} \right\}$ .

So, common elements in set$A$ and set$B$ are : $c,e$

So, we can say $A\cap B=\left\{ c,e \right\}$

Therefore, $A\cap B=\left\{ c,e \right\}$

Note: In all the examples above, the intersection is a subset of each set forming the intersection that is $A\cap B\subseteq A$ and$A\cap B\subseteq B$ . Two sets whose intersection is an empty set then they are called disjoint sets.