Definition:Cancellable Mapping
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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
- $f$ 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 $h_1 = h_2$.