Let the number of elements of the sets A and B be p and q, respectively. Then, the number relations from the set A to the set B is

Key Concepts Needed

1. Cartesian Product ($A \times B$)
   The set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.
   If $|A|=p$ and $|B|=q$, then
   $|A \times B| = p\, q$

2. Relation as a Subset
   A relation $R$ from $A$ to $B$ is any subset of $A \times B$.
   It can contain none, some, or all of the ordered pairs.

3. Counting Subsets
   For a set with $n$ elements, the number of all possible subsets is $2^{n}$ because each element can be kept or left out (2 choices) independently.

Chain of Thought to Solve

1. Find the total number of ordered pairs:
   $n = |A \times B| = p q$
2. Recognize that a relation is any subset of these $n$ pairs.
3. Use the subset-counting rule:
   $\text{Number of relations} = 2^{n} = 2^{p q}$

That’s why we get $2^{p q}$ relations.

What is the question?

You have two baskets of marbles:

• Basket A has p marbles.
• Basket B has q marbles.

A relation is like drawing arrows from some marbles in A to some marbles in B. You can choose to draw an arrow or not draw an arrow for each possible pair (any marble of A with any marble of B).

How many different ways can you decide which arrows exist?
(That’s the same as asking: How many relations are possible?)

How to think about it

1. First, count how many possible pairs (marble from A, marble from B) there are.
   If A has p marbles and B has q, then every marble in A can pair with every marble in B, so you get $p \times q$ possible pairs.
2. For each pair, you have two choices: either draw the arrow or don’t draw it.
3. Whenever you make several independent yes/no choices, multiply 2 by itself for each choice.
   So with $p \times q$ choices, you get $2^{p q}$ possible outcomes.

That’s it!

Step-by-Step Solution

1. Given $|A| = p$ and $|B| = q$.

2. Total ordered pairs in the Cartesian product:

   $|A \times B| = p q$

3. Each relation is a subset of $A \times B$.

4. Number of subsets of a set with $p q$ elements is $2^{p q}$.

$\boxed{\text{Number of relations from }A\text{ to }B = 2^{p q}}$