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

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 $A \times B$.