Mapping/Examples/Arbitrary Sets

Example of Relation which is not a Mapping
Let $A = \set {a, b, c}$.

Let $B = \set {1, 2, 3, 4}$.

Let $g_1 \subseteq {A \times B}$ such that:
 * $g_1 = \set {\tuple {a, 2}, \tuple {b, 4} }$

Let $g_2 \subseteq {A \times B}$ such that:
 * $g_2 = \set {\tuple {a, 3}, \tuple {b, 1}, \tuple {c, 2}, \tuple {c, 4} }$

Let $g_3 \subseteq {A \times B}$ such that:
 * $g_3 = \set {\tuple {a, 4}, \tuple {b, 4}, \tuple {c, 2} }$

Then $g_3$ is a mapping.

However, neither $g_1$ nor $g_2$ is a mapping.

Proof
We have that:
 * $\forall x \in A: \exists \tuple {x, y} \in g_3$


 * $\forall x \in A: \exists \tuple {x, y_1} \in g_3 \land \tuple {x, y_2} \in g_3 \implies y_1 = y_2$

and so $g_3$ is indeed a mapping.

We have:
 * $\nexists \tuple {x, y} \in g_1: x = c$

so $g_1$ is not a mapping.

We have:
 * $\tuple {c, 2} \in g_2$ and $\tuple {c, 4} \in g_2$

but:
 * $2 \ne 4$

and so $g_2$ is not a mapping.