Question

If $\alpha ,\xi ,\eta$ are non-empty sets then:
A. $\left( \alpha \times \beta \right)\bigcup \left( \xi \times \eta \right)=\left( \alpha \times \beta \right)\bigcap \left( \xi \times \eta \right)$
B. $\left( \alpha \times \beta \right)\bigcap \left( \xi \times \eta \right)=\left( \alpha \times \beta \right)\bigcap \left( \xi \times \eta \right)$
C. $\left( \alpha \bigcap \beta \right)\bigcup \left( \xi \bigcap \eta \right)=\left( \alpha \times \beta \right)\bigcup \left( \xi \times \eta \right)$
D. $\left( \alpha \bigcap \beta \right)=\left( \xi \bigcap \eta \right)=\left( \alpha \times \beta \right)\bigcup \left( \xi \times \eta \right)$

Answer 1

This type of question is based on set theory. We need to know what sets are and the different kinds of operations that can be performed on them. We solve this question by considering each equation one by one and checking to see if the left-hand side and the right-hand side match. If they do, then the obtained equation is marked as the answer.

Complete step by step solution:
For the given question, we need to know what sets are and the different operations performed on them. Sets are nothing but a collection of elements or things. We can group a number of things together. We call this grouped data as a set. The different kinds of set operations are as follows: Set Union, Set Intersection, Set Complement, Set Difference, Cartesian product of sets. These operations are usually performed on two sets (except for the complement of the set).
Set union of two sets A and B is nothing but a set containing elements that are present in both the sets A and B. This is represented as $A\bigcup B.$
Set intersection of two sets A and B is nothing but a set containing elements that are common in both the sets A and B. This is represented as $A\bigcap B.$
Set complement of a set A is a set containing all the elements that are present in the universal set except in both the set A. This is represented as $A'.$
Set difference of two sets A and B is defined as a set containing elements that are present only in the set A and not in set B. This is represented as $A-B.$
Cartesian product of two sets A and B is defined as a set containing elements that have an ordered pair of elements represented as $\left( {{a}_{n}},{{b}_{n}} \right)$ such that, ${{a}_{n}}\in A$ and ${{b}_{n}}\in B.$
We now consider the first option and we can note that there are two terms, $\left( \alpha \times \beta \right)$ which is the cartesian product of the terms $\alpha$ and $\beta ,$ $\left( \xi \times \eta \right)$ which is the cartesian product of $\xi$ and $\eta .$
Looking at the first equation, we can say that the intersection of the cartesian product of two sets can never be equal to the union of the cartesian product of two sets. Hence, option (A) is incorrect.
Considering the second equation, we can see both sides of the equation having the same terms. It is represented as the intersection of the cartesian product of two sets is equal to the intersection of the cartesian product of the exact same two sets. Here, left-hand side is equal to the right-hand side. Hence, option (B) is correct.
Considering the third option, we can note that the left-hand side of the equation consists of a set that has elements whereas the right-hand side of the equation consists of a set having the cartesian product of two sets taken with the union of the cartesian product of the other two sets. These two sides can never be equal. Therefore, left-hand side has terms represented as ${{a}_{1}},{{a}_{2}},\ldots {{b}_{1}},{{b}_{2}},\ldots \ldots$ whereas the right-hand side consists of elements $\left( {{a}_{1}},{{b}_{1}} \right),\left( {{a}_{2}},{{b}_{2}} \right)\ldots \ldots$ Hence, they can never be equal, and option (C) is incorrect.
Similar to the third option, the fourth option too has the left-hand side of the equation consisting of a set that has elements whereas the right-hand side of the equation consists of a set having the cartesian product of two sets taken with the union of the cartesian product of the other two sets. Hence, option (D) is incorrect.

So, the correct answer is “Option B”.

Note: To solving this question, the students need to have a good knowledge in the topic of sets and set operations. We can solve this question by another method, which is by using Venn diagrams. But in order to do so, the students need to know the set operations very well. This approach is simple too and effective to obtain the solution.