If$A=\left\{ 3,5,7,9,11 \right\}$ ,$B=\left\{ 7,9,11,13 \right\}$ ,$C=\left\{ 11,13,15 \right\}$ and$D=\left\{ 15,17 \right\}$ , find:
(i)$A\cap B$
(ii)$A\cap C$
(iii)$B\cap C$
(iv)$B\cap D$
(v)$B\cap \left( C\cup D \right)$
(vi)$A\cap \left( B\cup C \right)$

Hint: If two or more sets are given and we have to find the intersection of them, then we find all elements which are common in all of them. For example, if we have to find the intersection of set $A$ and set $B$ we find all the elements which are in set $A$ as well as in set $B$. It is represented by $A\cap B$ . Clearly, the intersection of two sets $A$ and set$B$ is a subset of both the sets that is $A\cap B\subseteq A$ and also, $A\cap B\subseteq B$. To find the union of two set$A$ and set$B$ we find all elements which are either in set $A$ or in set $B$ or in both the sets it is represented as$A\cup B$ .

Complete step-by-step answer:

(i) Now, to find$A\cap B$, where $A=\left\{ 3,5,7,9,11 \right\}$ and$B=\left\{ 7,9,11,13 \right\}$
 Now, we have to find the intersection of set $A$ and set $B$. So, we find all the common elements in both the sets.
So, common elements we can observe are: $\left\{ 7,9,11 \right\}$
So, $A\cap B=\left\{ 7,9,11 \right\}$

(ii) To find $A\cap C$, where $A=\left\{ 3,5,7,9,11 \right\}$ and $C=\left\{ 11,13,15 \right\}$
To, find $A\cap C$ we follow the same procedure as in part (i) of the question, that is we find common elements in the set $A$ and set $C$.
So, common elements we observe are:$\left\{ 11 \right\}$
So, $A\cap C=\left\{ 11 \right\}$

(iii) To find $B\cap C$ , where $B=\left\{ 7,9,11,13 \right\}$ and $C=\left\{ 11,13,15 \right\}$
So, to find $B\cap C$, we follow the same procedure as in part (i) and part (ii), we simply find the common elements in set $B$ and set $C$.
So, common elements we observe are:$\left\{ 11,13 \right\}$
So,$B\cap C=\left\{ 11,13 \right\}$

(iv) To find$B\cap D$ , where$B=\left\{ 7,9,11,13 \right\}$ and$D=\left\{ 15,17 \right\}$
Now, to find$B\cap D$ ,we find the common elements which are in set $B$ as well as in set $D$ But as we can see there are no common terms in set $B$ and set $D$.
So, we write$B\cap D=\left\{ \Phi \right\}$ .
This means that set $B$ and set $D$ are disjoint sets.

(v) To find$B\cap \left( C\cup D \right)$, where$B=\left\{ 7,9,11,13 \right\}$, $C=\left\{ 11,13,15 \right\}$ and $D=\left\{ 15,17 \right\}$
So, here we have to first find the union of set $C$ and set $D$ $(C\cup D)$ and then we have to find the intersection of set $B$ with the first founded set $(C\cup D)$ , that is $B\cap \left( C\cup D \right)$
Now, to find set $(C\cup D)$, we find all elements which are either in set $C$ or in set $D$ or both the set.
So, $C\cup D=\left\{ 11,13,15,17 \right\}$
Now, to find $B\cap \left( C\cup D \right)$, we find all the elements which are common in set $B$ and set$(C\cup D)$, which are:$\left\{ 11,13 \right\}$
So, $B\cap \left( C\cup D \right)=\left\{ 11,13 \right\}$ .

(vi) To find $A\cap \left( B\cup C \right)$ ,where $A=\left\{ 3,5,7,9,11 \right\}$, $B=\left\{ 7,9,11,13 \right\}$ and $C=\left\{ 11,13,15 \right\}$
We follow the same procedure as in part (v) , that is we first find set $B$ union set $C$ $\left( B\cup C \right)$ and then, we find intersection of set $A$ with set $\left( B\cup C \right)$.
Now, for$\left( B\cup C \right)$, we find the elements which are either in set $B$ or set $C$ or in both the set.
So, $B\cup C=\left\{ 7,9,11,13,15 \right\}$
Now, we find elements common in set $A$ and set $\left( B\cup C \right)$, which are:$\left\{ 7,9,11 \right\}$
So, $A\cap \left( B\cup C \right)=\left\{ 7,9,11 \right\}$

 Note: If we find the intersection of two set, the resulting set is the subset of the former set. Intersection of two sets can be $\Phi $ that means they are disjoint sets.