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 $B\cap C$.
$\begin{aligned} & \Rightarrow A\times \left( B\cap C \right)=\\{2,3\\}\times \\{5\\} \\\ & \Rightarrow A\times \left( B\cap C \right)=\\{\left( 2,5 \right),\left( 3,5 \right)\\} \\\ \end{aligned}$
(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.