Question

If the set A has p elements, \[B\] has \[q\] elements then the number of elements in \[A \times B\] is
\[1)p + q\]
\[2)p + q + 1\]
\[3)pq\]
$4){p^2}$

Answer 1

Hint: We need to find the number of elements in \[A \times B\;\]. We solve this question by using the concept of relations between two sets . We find the Cartesian product between \[A\] and \[B\] to find the number of elements in \[A \times B\;\]. From the relation set we can conclude the number of elements in the relation set \[A \times B\].

Complete step-by-step solution:
Given :
\[\;A\] has \[p\] elements, and \[B\] has \[q\] elements
For the number of elements in \[A \times B\], we have to find the relation from $A$ to $B$ , so we have to find the cartesian product of the sets in the manner of \[A \times B\].
Let us consider if there are two sets $P$ and $Q$ such that \[P = \left\{ {1,2} \right\}\] and \[Q = \left\{ {a,b} \right\}\], then the relation between $P$ and $Q$ from $P$ to $Q$ is given as :
\[P \times Q = \left\{ {\left( {p,q} \right):p \in P,q \in Q} \right\}\]
I.e. the relation from \[P\] to \[Q\] would be ,
\[P \times Q = \left\{ {\left( {1,a} \right),\left( {1,b} \right),\left( {2,a} \right),\left( {2,b} \right)} \right\}\]
he number of elements in \[P \times Q\] is $4$
Also , the formula for the number of elements in \[A \times B\] would be equal to the product of the number of elements in the two sets
So ,
the number of elements in \[A \times B = \] number of elements of \[A \times \] number of elements of \[\;B\]
the number of elements in \[A \times B = p \times q\]
Thus the number of elements in the relation \[A \times B\] is \[pq\].
Hence , the correct option is \[\left( 3 \right)\;\].

Note: If either $P$or $Q$ given set is an empty set , then \[P \times Q\] will also be an empty set .
In general \[A \times B \ne B \times A\]
A relation R from a set A to a set B is a subset of the Cartesian product \[A \times B\] obtained by describing a relationship between the first element x and the second element y of the ordered pairs in \[A \times B\].