How is this a function? - Analysis. 
Let $X = \{1, 2, 3\}, Y = \{4, 5, 6\}$. Define $F \subseteq X \times Y$ as
$F = \{(1, 4),(2, 5),(3, 5)\}$.
Then $F$ is a function.

I simply do not see how this could be a function, as there is nothing that it is mapping to, if anyone can explain how this is a function, that would be lovely.

*

*In a positive way, a set $S$ is a function if and only if : $S$ is a binary relation ( that is, a set such that all its elements are ordered pairs) AND for all objects $x$, $y$ and $z$ the pairs $(x,y)$ and $(x,z)$ belong to $S$ only if $y=z$ ( meaning, only if these " two" pairs are , in fact, identical). In short, quoting Enderton : " a function is a single valued relation" .


*This test / definition can be turned negatively using DeMorgan's law. Let $S$ be a set ; there are only $2$ reasons you can give in order to assert that $ S$ is not a function ( and each reason is sufficient by itself)  :

(1) $S$ is not a set of ordered pairs , that is $S$ is not a binary
relation

OR

(2) there are at least, in $S$ ,  two different ordered pairs with the
same first element (meaning : two pairs with the same first element,
but with non-identical second elements).


*

*Applying this to the case you present :

(1) is there an element of $F$ that is not an ordered pair?
(2) are there in $F$ two pairs with the same first element, but with different second elements?

*

*By reason (1) the set $\{(a, 1), c, (b,2)\}$ is not a function.


*By reason (2) the set $\{(a, 1), (b,3), (b,2)\}$ is not a function.
