Let A = {1, 2} and B = {3, 4}. Find the number of relations from A to B. 16 32 64 108

1. Break Down the Problem

To find the number of relations from set $A$ to set $B$, we need to understand what a relation is. A relation from set $A$ to set $B$ is essentially a subset of the Cartesian product $A \times B$.

2. Relevant Concepts

The Cartesian product $A \times B$ consists of all possible ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. The number of elements in $A \times B$ can be calculated using the formula: $|A \times B| = |A| \times |B|$ where $|A|$ and $|B|$ are the number of elements in sets $A$ and $B$, respectively.

3. Analysis and Detail

1. Calculate the size of set $A$: $|A| = 2$
2. Calculate the size of set $B$: $|B| = 2$
3. Calculate the Cartesian product: $|A \times B| = |A| \times |B| = 2 \times 2 = 4$ The pairs in the Cartesian product $A \times B$ are: $\{(1, 3), (1, 4), (2, 3), (2, 4)\}$
4. A relation from $A$ to $B$ is any subset of this Cartesian product. The number of subsets of a set with $n$ elements is given by $2^n$.

4. Verify and Summarize

Since $|A \times B| = 4$, the number of subsets (relations) is: $2^{|A \times B|} = 2^4 = 16$

Final Answer

The number of relations from $A$ to $B$ is $\boxed{16}$.