Definition:Cancellable Mapping

Definition

Let $X$ and $Y$ be sets.

Let $f: X \to Y$ be a mapping from a $X$ to $Y$.

Then

$f$ is a cancellable mapping

if and only if:

$f$ is both a left cancellable mapping and a right cancellable mapping.

Left Cancellable Mapping

A mapping $f: Y \to Z$ is left cancellable (or left-cancellable) if and only if:

$\forall X: \forall \struct {g_1, g_2: X \to Y}: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

That is, for any set $X$, if $g_1$ and $g_2$ are mappings from $X$ to $Y$:

If $f \circ g_1 = f \circ g_2$

then $g_1 = g_2$.

Right Cancellable Mapping

A mapping $f: X \to Y$ is right cancellable (or right-cancellable) if and only if:

$\forall Z: \forall \paren {h_1, h_2: Y \to Z}: h_1 \circ f = h_2 \circ f \implies h_1 = h_2$

That is, if and only if for any set $Z$:

If $h_1$ and $h_2$ are mappings from $Y$ to $Z$

then $h_1 \circ f = h_2 \circ f$ implies $h_1 = h_2$.