Question

If $M \cup N = N \cup R$ and $M \cap N = N \cap R$ then which of the following is necessarily true?
A. $M = N$
B. $N = R$
C. $M = R$
D. $M = N = R$

Answer 1

Hint: Here we need to choose the option that suits the given $M \cup N = N \cup R$ and $M \cap N = N \cap R$. We need to apply the important property of union and intersection to obtain the desired answer. The union of the two given sets and the intersection of the sets are commutative. Using the commutative property, we can choose the correct option.

Complete step-by-step answer:
We are given $M \cup N = N \cup R$ and $M \cap N = N \cap R$
Before getting into the solution, we shall consider an example.
Let $A = \left\{ {1,2,3,5} \right\}$ and $B = \left\{ {2,4,5,6,7} \right\}$
Now, we shall calculate the union of the above two sets.
To find the union of the sets, we need to combine all the elements of both sets.
Thus, we have $A \cup B = \left\{ {1,2,3,4,5,6,7} \right\}$
Now, we shall find the commutative of the union of the above two sets (i.e. we need to obtain $B \cup C$ )
Hence, we get $B \cup C = \left\{ {1,2,3,4,5,6,7} \right\}$
We can note that $A \cup B = B \cup A$
That is, the commutative property holds for the union of two sets.
Similarly, we shall calculate the intersection of the above two sets.
To find the intersection of the sets, we need to collect the common elements from the sets.
Thus, we have $A \cap B = \left\{ {2,5} \right\}$
Now, we shall find the commutative of the intersection of the above two sets (i.e. we need to obtain $B \cap C$ )
Hence, we get $B \cap C = \left\{ {2,5} \right\}$
We can note that $A \cap B = B \cap A$
That is, the commutative property holds for the intersection of two sets.
Hence, we can able to say that the union of the sets and the intersection of the sets is commutative.
Now, we shall get into the solution.
It is given that $M \cup N = N \cup R$
Union is operated on the sets M and N (i.e. $M \cup N$ )
The union of sets M and N is commutative
Thus, $M \cup N = N \cup M$
Since we have $M \cup N = N \cup R$and we found $M \cup N = N \cup M$, we can able to conclude that $M = R$
Similarly, the intersection is operated on the sets M and N (i.e.$M \cap N$ )
The intersection of the sets M and N is commutative.
Thus, $M \cap N = N \cap M$
Since we have $M \cap N = N \cap R$and we found $M \cap N = N \cap M$, we can able to conclude that $M = R$
Therefore, $M = R$and option C is the correct answer.

So, the correct answer is “Option C”.

Note: Note that the values of A union B and B union A are the same, also for an intersection B and B intersection A is the same. But for $A - B$(A difference B) and $B - A$(B difference A) are not the same. When the result is not changed though we change the order is said to be commutative.