Question

If we are given elements of two sets as n(A) = m and n(B) = n, then \[n\left( A\times B \right)=mn\]. What is the value of \[n\left( A\times B \right)=mn\], if m = 6 and n = 8.

Answer 1

Hint: First of all, try to recollect the meaning of n(A), n(B), and \[n\left( A\times B \right)\]. Now use the formula for \[n\left( A\times B \right)=mn\] and substitute in it the values of m and n to get the required answer.

Complete step-by-step answer:
We are given that if n(A) = m and n(B) = n and \[n\left( A\times B \right)=mn\]. Then we have to find the value of \[n\left( A\times B \right)\] if m = 6 and n = 8. Before proceeding with the question, let us understand a few terms.
Sets: We can say that a set is a well-defined collection of distinct objects. For example {1, 3, 4, 6, 12} is a set containing 5 elements. Also, we have sets like a set of even numbers, prime numbers, etc. The number of elements in any set, say set A is represented by n(A). We can have an empty set as well.
Cartesian Product of Sets: The Cartesian products of sets means the product of two non-empty sets in an ordered way. Or, we can say that it is the collection of all ordered pairs obtained by the product of two non-empty sets. An ordered pair means that two elements are taken from each set. The Cartesian product of two non-empty sets A and B is denoted by \[A\times B\]. So,
\[A\times B=\left\{ \left( a,b \right):a\in A,b\in B \right\}\]
Number of elements in set \[\left( A\times B \right)\] = (Number of elements in any set A) \[\times \] (Number of elements in any set B)
\[n\left( A\times B \right)=n\left( A \right)\times n\left( B \right)\]
Now let us consider our question. Here we are given that, n(A) = m and n(B) = n. Also, we know that,
\[n\left( A\times B \right)=n\left( A \right)\times n\left( B \right)=mn....\left( i \right)\]
Now, we are given that m = 6 and n = 8. So, by substituting m = 6 and n = 8 in equation (i), we get,
\[n\left( A\times B \right)=mn=6\times 8\]
So we get,
\[n\left( A\times B \right)=48\]

Note: Students must note that just applying the formula that is already given in this question and getting to answer is not important but they must understand the meaning of each term. For example, here it is very important to learn the concept of the Cartesian product of two sets as it will come handy in doing other questions of this topic when formula would not be mentioned in questions.