I'm given 2 sets $G = \{g_1, g_2, ..., g_m\}$ and $R = \{r_1, r_2, ..., r_n\}$. Now I have to define the set of all mappings of $G$ to $R$. I did it like this: $M = \{(K, L) | K \subseteq G, L \subseteq R, ???\} $ But I'm stuck at the ???. How can I write this mapping formally?
Formally write this mapping from one set to another
-3
$\begingroup$
functions
elementary-set-theory
1 Answers
1
Your $M$ is mistaken because it is composed of pairs of subsets of $G$ and, respectively, $R$. The correct version is
$$M = \{ f \subseteq G \times R \mid \forall g \in G \ \exists ! r \in R \ \text{such that } (g,r) \in f \} $$
(i.e. no element $g \in G$ may be mapped into several distinct elements of $R$ - that would still be a binary relation, but no longer a map).