Considering any two non empty sets ‘A’ & ‘B’ verify that n(AxB)=pq and the total no.of relations 
from A to B is 2pq where n(A)=p and n(B)=q.

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 \( 2^{pq} \).

### Do you want more details or have any

Considering any two non-empty sets ‘A’ & ‘B’ verify that n(AxB)=pq and the total number of relations from A to B is 2pq where n(A)=p and n(B)=q.

This problem demonstrates how to verify that the number of elements in the Cartesian product of two non-empty sets A and B is pq and explores how to find the total number of relations from A to B as 2^(pq). Key concepts include set theory, Cartesian products, and relations, suitable for Grades 10-12 students.

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