List all relations $\{a,b\} \to \{c,d\}$, assuming $a \neq b$ and $c \neq d$. Which of them are maps?
So I know the cartesian product gives $\{(a,c),(a,d),(b,c),(b,d)\}$.
And the relations will be subsets of this cartesian product:
$1) \emptyset,\\ 2)\{(a,c)\},\\ 3)\{(a,d)\},\\ 4)\{(b,c)\},\\ 5)\{(b,d)\},\\ 6)\{(a,c),(a,d)\},\\ 7)\{(a,c),(b,c)\},\\ 8)\{(a,c),(b,d)\},\\ 9)\{(a,d),(b,c)\},\\ 10)\{(a,d),(b,d)\},\\ 11)\{(b,c),(b,d)\},\\ 12)\{(a,c),(a,d),(b,c)\},\\ 13)\{(a,c),(a,d),(b,d)\},\\ 14)\{(a,c),(b,c),(b,d)\},\\ 15)\{(a,d),(b,c),(b,d)\},\\ 16)\{(a,c),(a,d),(b,c),(b,d)\}. $
Now for maps, what do I need to look for specifically? Why is the empty set or set (2) just below it not a map? When I look at relation (2), I see an element where the first component is from A = {a,b}, the second component is from B = {c,d} and the first component maps to a unique second component. I know I'm wrong, but why?
Following that, why is relation (12) not a map?
And lastly, why is relation (6) not a map?
Relation $R\subseteq X\times Y$ is a map if and only for every $x\in X$ there is exactly one $y\in Y$ such that $\langle x,y\rangle\in R$.
Here $X=\{a,b\}$ and $Y=\{c,d\}$ with $a\neq b$ and $c\neq d$.
So in this context $R$ is a map if it is of the form $\{\langle a,y\rangle,\langle b,z\rangle\}$ with $y,z\in\{c,d\}$.
It is not empty (relation 1 is not a map).
It must contain one of the pairs (not both) $\langle b,c\rangle$ and $\langle b,d\rangle$ (relation 2 is not a map).