Question

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C= {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find (i) $\left( A\cap B \right)\cap \left( B\cap C \right)$ (ii) $\left( A\cup D \right)\cap \left( B\cup C \right)$

Answer 1

Hint:The union of two sets A and B, written $A\cup B$, is the combination of the two sets. The intersection of two sets A and B, written $A\cap B$, is the common element of the two sets.

Complete step-by-step answer:
The symbol used for the union of two sets is $\cup $ .
Therefore, symbolically, we write union of the two sets A and B is $A\cup B$ which means A union B.
Therefore, \[A\cup B\text{ }=\text{ }\{x\text{ }:\text{ }x\in A\text{ }or\text{ }x\in B\}\]
The symbol used for the intersection of two sets is $\cap $.
Therefore, symbolically, we write the intersection of the two sets A and B is $A\cap B$ which means A intersection B.
The intersection of two sets A and B is represented as \[A~\cap B\text{ }=\text{ }\{x\text{ }:\text{ }x\in A\text{ }and\text{ }x\in B\}~\]
The given sets are
A = {1, 2, 3, 4, 5}
B = {4, 5, 6, 7, 8}
C = {7, 8, 9, 10, 11}
D = {10, 11, 12, 13, 14}
(i) $A\cap B=\{4,5\}$ and $B\cap C=\{7,8\}$
The intersection of two sets $A\cap B$ and $B\cap C$ is the set of elements which are common in $A\cap B$ and $B\cap C$.
$\left( A\cap B \right)\cap \left( B\cap C \right)=\phi $= Empty set

(ii) $A\cup D=\{1,2,3,4,5,10,11,12,13,14\}$ and $B\cup C=\{4,5,6,7,8,9,10,11\}$ .The intersection of two sets $A\cup D$ and $B\cup C$ is the set of elements which are common in $A\cup D$ and $B\cup C$.
$\left( A\cup D \right)\cap \left( B\cup C \right)=\{4,5,10,11\}$

Note: The set X and set Y are the subsets of $X\cup Y\text{ or X}\cap \text{Y}$. The union or intersection of two sets is commutative. The union or intersection operations are performed when the sets are expressed in roster form. The union or intersection of any set with the empty set is always the set itself.