Question

Given $M = (0,1,2)$ and $N = (1,2,3)$. Find $(N - M) \times (N \cap M)$.
(A) $\\{ (3,1),\;(3,2)\\}$
(B) $\\{ (3, - 1),\;(3, - 2)\\}$
(C) $\\{ ( - 3, - 1),\;( - 3,2)\\}$
(D) $\\{ (3, - 1),\;( - 3, - 2)\\}$

Answer 1

Here we are given two different sets $M$ and $N$. We need to know the different operations we can perform on sets. The operations are represented by various symbols, remember that the symbol ‘$-$’ represents the operation ‘difference’, symbol ‘$\times$’ represents the operation ‘cross product or Cartesian product’ and the symbol ‘$\cap$’ represents the operation intersection.

Complete step by step solution:
The sets given to us are:
$M = (0,1,2)$, $N = (1,2,3)$
The problem we need to solve is: $(N - M) \times (N \cap M)$
We approach this question by first solving the operations that are within brackets then move on to the operation outside the bracket.
Let us take the first bracket;
Here we see that the operation is ‘$-$’ that is ‘difference between sets $N$ and $M$’. This just means that we need to make another set $N - M$ that contains the elements of the set $N$ but only those elements of $N$ that are not there in the set $M$.
So the set $N - M = \\{ 3\\}$
Now taking the second bracket we have;
The operation ‘$\cap$’ is being performed, this means that we need to find ‘$N$ intersection $M$’. This just means that the set $N \cap M$ will have all the elements that are there in common in those sets $N,\;M$.
It implies that $N \cap M = \\{ 1,2\\}$
Now we can go ahead with operating the sets $N - M = \\{ 3\\}$ and $N \cap M = \\{ 1,2\\}$ with the operation ‘$\times$’.
The ‘$\times$’ operation takes the cross product between two sets, here this operation is between $N - M$ and $N \cap M$.
When the cross product of two sets are to be found, it means that the new set $(N - M) \times (N \cap M)$ contains ordered pairs. So each value of within the set $(N - M) \times (N \cap M)$ will be an ordered pair. We know that every ordered pair we make has to have an x-coordinate and y-coordinate.
Form this ordered pair in this way:
The x-coordinate value is an element from the first set here that is from $N - M$.
Similarly the y-coordinate value is an element from the second set here that is from $N \cap M$.
To verify if the final answer is right, just remember that the number of elements in the final cross product must contain is the product of number of elements in first set and number of elements in second set. Here the total number within $(N - M) \times (N \cap M)$ will be; $\Rightarrow n(N - M) \times n(N \cap M) = 1 \times 2$
$\Rightarrow n(N - M) \times n(N \cap M) = 2$
Now we proceed to finding the cross products according to the above mentioned format:
Only two elements will be there in the cross product set:
The first element will be $(3,1)$, second element will be $(3,2)$.

Hence the correct answer is set $\\{ (3,1),\;(3,2)\\}$ so now if we look at the given options:

Option (A) $\\{ (3,1),\;(3,2)\\}$is correct. This is because all the elements of the cross product are there within either of the sets $(N - M)$ or $(N \cap M)$. The elements are placed correctly also.
Option (B) $\\{ (3, - 1),\;(3, - 2)\\}$is incorrect. Here there are values $- 1,\; - 2$ within the cross product but they are not there in the sets $(N - M)$ or $(N \cap M)$. So this is a wrong answer.
Option (C) $\\{ ( - 3, - 1),\;( - 3,2)\\}$ is incorrect. Here the values $- 1,\; - 3$ are present within the cross product but they are not there in the sets $(N - M)$ or $(N \cap M)$, this is not a valid cross product. So this is a wrong answer.
Option (D) $\\{ (3, - 1),\;( - 3, - 2)\\}$ is incorrect. Here the values $- 1,\; - 2,\; - 3$ are present within the cross product but they are not there in any of the sets that are being crossed that is sets $(N - M)$ or $(N \cap M)$, so this is not a valid cross product. So this is a wrong answer.
In general the individual sets do not have any negative values but in each of the cross products in the some options there are negative numbers. It is not possible because only elements from the individual sets will belong to any cross product. So we can classify those options as incorrect answers.
Therefore the correct answer is in option (A) $\\{ (3,1),\;(3,2)\\}$.

Note:
There are some more operations that we can perform on sets. If we take two sets say $A,\;B$ and perform the operation ‘$A \cup B$’ then the symbol ‘$\cup$’ represents the operation ‘union’ where all the elements of set $A,\;B$ will be combined. Then if we perform an operation ‘${A^C}$’ on the set $A$, the symbol ‘$^C$’ represents the complement operation where the formed set will have every element that is there within a universal set but does not belong within $A$.