Question

If $ n(A) = p $ and $ n(B) = q $, then the number of relations from the set $ A $ to the set $ B $ is

• A. \( 2^{p+q} \)
• B. \( 2^{pq} \)
• C. \( p + q \)
• D. \( pq \)

Hint:

The number of relations from set \( A \) to set \( B \) is determined by the power of 2, raised to the product of the number of elements in each set. This represents all possible subsets of the Cartesian product \( A \times B \).

Answer 1

The Correct Option is B

To determine the number of possible relations between two finite sets, we analyze their Cartesian product and subsets.

1. Given Information:
- Set A has $p$ elements: $n(A) = p$
- Set B has $q$ elements: $n(B) = q$

2. Cartesian Product:
The Cartesian product $A × B$ contains all ordered pairs:
$ A × B = \{(a,b) \mid a \in A, b \in B\} $
Number of elements in $A × B$ is:
$ n(A × B) = p × q = pq $

3. Subsets of Cartesian Product:
For any set with $k$ elements, there are $2^k$ possible subsets.
Thus, $A × B$ has:
$ 2^{pq} $ possible subsets

4. Relations as Subsets:
Every relation from A to B corresponds to a subset of $A × B$.
Therefore, the number of possible relations equals the number of subsets.

Final Result:
The total number of relations from set A to set B is: $ 2^{pq} $

Answer 2

To determine the number of relations from set \( A \) to set \( B \), we need to understand the concept of relations between two sets.

Given sets \( A \) and \( B \) with cardinalities \( n(A)=p \) and \( n(B)=q \), a relation from \( A \) to \( B \) is a subset of the Cartesian product \( A \times B \).

The Cartesian product \( A \times B \) consists of all possible ordered pairs \((a,b)\) where \( a \in A \) and \( b \in B \). The total number of such ordered pairs is given by the product \( p \times q \) or simply \( pq \).

Each element of this Cartesian product can either be in a subset or not. Hence, for each of the \( pq \) elements, there are 2 choices (include it or exclude it in a relation).

Therefore, the total number of possible subsets (relations) is \( 2^{pq} \).

Thus, the number of relations from set \( A \) to set \( B \) is \( 2^{pq} \).