The question you're asking is about how many relations are possible between two sets.

Let:

• Set $A$ have 3 elements.
• Set $B$ have 2 elements.

Concept:

A relation from set $A$ to set $B$ is any subset of the Cartesian product $A \times B$. The number of elements in $A \times B$ is given by the product of the number of elements in $A$ and $B$.

$\text{Number of elements in } A \times B = |A| \times |B| = 3 \times 2 = 6$

For each pair in $A \times B$, there are 2 possibilities: either the pair is included in the relation, or it is not included.

Therefore, the total number of possible relations is: $2^{|A \times B|} = 2^6 = 64$

Answer:

The correct option is d) 64.

Would you like further details, or have any questions?

Here are five related questions to expand on this topic:

1. What is the definition of a Cartesian product of two sets?
2. How can we represent a relation from set $A$ to set $B$ graphically?
3. What is the difference between a relation and a function?
4. How would the number of possible relations change if both sets $A$ and $B$ had the same number of elements?
5. Can a relation from set $A$ to set $B$ be a one-to-one relation?

Tip: The number of relations depends on the number of pairs in the Cartesian product of the two sets. Always remember this when calculating possible relations.