Let $A=\{1{,}2\}$ and $B=\{3{,}4\}$. Find the number of relations from $A$ to $B$.

Step 1: Determine the number of elements in sets A and B

First, we need to find the number of elements in each set. Set $A$ has two elements, $1$ and $2$. Set $B$ also has two elements, $3$ and $4$. We denote the number of elements in a set $S$ as $|S|$.

Step 2: Find the Cartesian product A x B

The Cartesian product $A \times B$ is the set of all possible ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. We list all such pairs.

Step 3: Determine the number of elements in the Cartesian product

The number of elements in the Cartesian product $A \times B$ is the product of the number of elements in set $A$ and the number of elements in set $B$. In this case, it is $2 \times 2 = 4$.

Step 4: Calculate the number of relations from A to B

A relation from set $A$ to set $B$ is any subset of the Cartesian product $A \times B$. If a set has $n$ elements, then the number of its subsets is $2^n$. Since $|A \times B| = 4$, the number of relations from $A$ to $B$ is $2^4 = 16$.

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