Why are $\{(1,3)\}$ and $\{(2,4)\}$ not functions from $\{1,2\}$ to $\{3,4\}$?

Question

Consider set $A = \{1, 2\}$ and set $B = \{3, 4\}$, so $A\times B = \{(1,3),(1,4),(2,3),(2,4)\}$.

We know that any subset of $A\times B$ is a relation for example, $R_1 = \{(1,3)\}$, $R_2 = \{(2,4)\}$ and $R_3 = \{(1,3),(2,4)\}$. We also know that any relation is a function in which there is no repetition in domain and that domain is a subset of set $A$. Why can't $R_1$ and $R_2$ be functions? If they are, then why there are only $2^2$ functions possible from $A$ to $B$?

Answer 1

A function from $A$ to $B$ must have the entirety of $A$ as its domain, not just a subset. So, for instance, while you can consider $R_1$ a function on the set $\{1\}$, it is not a function from $A$ to $B$, since $R_1(2)$ has no value.

Answer 2

A function from $A$ to $B$ is any $F\subset A\times B$ such that for EVERY $x\in A$ there exists exactly one $y\in B$ with $(x,y)\in F.$

In your Q , $R1$ and $R2$ are functions. They just aren't functions from $A$ to $B.$