Question

Consider three sets A, B, C such that A = {2, 3}, B = {4, 5} and C = {5, 6}. Find,
(i) $A\times \left( B\cap C \right)$
(ii) $\left( A\times B \right)\cup \left( B\times C \right)$

Answer 1

Hint: For any two sets A and B, the intersection of the two sets A and B (denoted by $A\cap B$) can be found by forming a set which contains the common elements of set A and set B. The cross product of two sets A and B (denoted by $A\times B$) is given by pairing each element of set A with each element of set B. Also, the union of the two sets A and B (denoted by $A\cup B$) is the set of elements which are in set A, in set B, or in both set A and set B. Using this, we can solve this question.

Complete step by step solution:
In the question, we are given three sets A, B, C such that A = {2, 3}, B = {4, 5} and C = {5, 6}.
(i) In this part, we are required to find $A\times \left( B\cap C \right)$.
The intersection of the two sets B and C (denoted by \[B\cap C\]) can be found by forming a set which contains the common elements of set B and set C. So, we can say,
\[B\cap C\] = {5}
The cross product of two sets A and \[B\cap C\] (denoted by $A\times \left( B\cap C \right)$) is given by pairing each element of set A with each element of set \[B\cap C\].
$\begin{align}
  & \Rightarrow A\times \left( B\cap C \right)=\{2,3\}\times \{5\} \\
 & \Rightarrow A\times \left( B\cap C \right)=\{\left( 2,5 \right),\left( 3,5 \right)\} \\
\end{align}$
(ii) In this part, we are required to find $\left( A\times B \right)\cup \left( B\times C \right)$.
The cross product of two sets A and B (denoted by $A\times B$) is given by pairing each element of set A with each element of set B.
$\Rightarrow A\times B=\{\left( 2,4 \right),\left( 2,5 \right),\left( 3,4 \right),\left( 3,5 \right)\}$
Similarly, we can find $B\times C=\left\{ \left( 4,5 \right),\left( 4,6 \right),\left( 5,5 \right),\left( 5,6 \right) \right\}$.
We have to find the union of the above to obtain cross products which are given by forming a set of elements in set $A\times B$, or in set $B\times C$, or in both the sets.
So, $\left( A\times B \right)\cup \left( B\times C \right)=\left\{ \left( 2,4 \right),\left( 2,5 \right),\left( 3,4 \right),\left( 3,5 \right),\left( 4,5 \right),\left( 4,6 \right),\left( 5,5 \right),\left( 5,6 \right) \right\}$.

Note: It is an easy question which can be done by the basic knowledge of set theory. The only possibility of error in this question is that if one misreads the question. There is a possibility that one may read the union sign as intersection sign or vice versa which may lead us to an incorrect answer.