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# If we have the sets as A = {2,4} and B = {3,4,5} then $\left( A\cap B \right)\times \left( A\cup B \right)$ is\begin{align} & A.\left\{ \left( 2,4 \right),\left( 3,4 \right),\left( 4,2 \right),\left( 5,4 \right) \right\} \\ & B.\left\{ \left( 2,3 \right),\left( 4,3 \right),\left( 4,5 \right) \right\} \\ & C.\left\{ \left( 2,4 \right),\left( 3,4 \right),\left( 4,4 \right),\left( 4,5 \right) \right\} \\ & D.\left\{ \left( 4,2 \right),\left( 4,3 \right),\left( 4,4 \right),\left( 4,5 \right) \right\} \\ \end{align}

Last updated date: 23rd Jun 2024
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Hint: In this question, we are given two sets A and B. We need to find $\left( A\cap B \right)\times \left( A\cup B \right)$. For this, we will first $\left( A\cap B \right)$ by taking elements from set A and B which are present in both sets. Then we will find $\left( A\cup B \right)$ where we will take all elements of set A and set B without writing repeated elements twice. At last we will find the set having ordered pairs of elements in the form (x,y) where x is the element which belongs to $\left( A\cap B \right)$ and y is the element which belongs to $\left( A\cup B \right)$.

Complete step-by-step solution:
Here we are given the set A and set B as: A = {2,4} and B = {3,4,5}
We need to find the set, $\left( A\cap B \right)\times \left( A\cup B \right)$.
For this let us first find the elements that will belong to $\left( A\cap B \right)$. As we know, $\left( A\cap B \right)$ means the intersection of A and B i.e. we need to find the elements which are common in both the sets. By observing we can see that only element 4 lies in both these sets. Therefore, 4 lies in the intersection of A and B we get, $\left( A\cap B \right)=\left\{ 4 \right\}$.
Now let us find the elements that will belong to $\left( A\cup B \right)$ i.e. we need to find the elements from both these sets. We will be writing all the elements from A and B without writing repeated elements twice. So we have A union B as, $\left( A\cup B \right)=\left\{ 2,3,4,5 \right\}$ (4 was repeated but written once only).
Now we need to find the Cartesian product of $\left( A\cap B \right)$ and $\left( A\cup B \right)$ i.e. $\left( A\cap B \right)\times \left( A\cup B \right)$.
We know Cartesian products of any two sets P and Q are written in the form (x,y) where $x\in P\text{ and }y\in Q$.
Therefore, $\left\{ 4 \right\}\times \left\{ 2,3,4,5 \right\}=\left\{ \left( 4,2 \right),\left( 4,3 \right),\left( 4,4 \right),\left( 4,5 \right) \right\}$.
Therefore, $\left( A\cap B \right)\times \left( A\cup B \right)=\left\{ \left( 4,2 \right),\left( 4,3 \right),\left( 4,4 \right),\left( 4,5 \right) \right\}$.
Hence option D is the correct answer.

Note: Students should note that $A\times B$ is not equal to $B\times A$. They should not try to find $\left( A\cup B \right)\times \left( A\cap B \right)$ instead of $\left( A\cap B \right)\times \left( A\cup B \right)$. They should not get confused between $\cup \text{ and }\cap$. Here $\cup$ represent the intersection of sets and $\cap$ represent the union of sets.