We are given two non-empty sets $A$ and $B$, where the number of elements in set $A$ is denoted by $p$ (i.e., $n(A) = p$) and the number of elements in set $B$ is denoted by $q$ (i.e., $n(B) = q$).

Step 1: Verifying that $n(A \times B) = pq$

The Cartesian product $A \times B$ is defined as the set of all ordered pairs $(a, b)$, where $a \in A$ and $b \in B$. Thus, for each element in set $A$, we can pair it with every element in set $B$, forming a pair $(a, b)$.

Since there are $p$ elements in set $A$ and $q$ elements in set $B$, the total number of possible pairs $(a, b)$ is the product of the number of elements in $A$ and $B$, which is: $n(A \times B) = p \times q = pq$ Thus, the number of elements in $A \times B$ is indeed $pq$.

Step 2: Finding the total number of relations from $A$ to $B$

A relation from set $A$ to set $B$ is defined as a subset of the Cartesian product $A \times B$. Since $A \times B$ contains $pq$ elements (as shown above), the number of subsets of $A \times B$ is equal to $2^{pq}$, because for each element in $A \times B$, we have two choices: either to include it in the relation or not.

Therefore, the total number of relations from $A$ to $B$ is given by: $\text{Total number of relations} = 2^{pq}$

Conclusion:

• The number of elements in $A \times B$ is $pq$.
• The total number of relations from $A$ to $B$ is $2^{pq}$.

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