Editing Equaliser (mathematics) (section)

== Definitions ==

Let ''X'' and ''Y'' be [[Set (mathematics)|sets]].
Let ''f'' and ''g'' be [[function (mathematics)|function]]s, both from ''X'' to ''Y''.
Then the ''equaliser'' of ''f'' and ''g'' is the set of elements ''x'' of ''X'' such that ''f''(''x'') equals ''g''(''x'') in ''Y''.
Symbolically:
: <math> \operatorname{Eq}(f, g) := \{x \in X \mid f(x) = g(x)\}. </math>
The equaliser may be denoted Eq(''f'', ''g'') or a variation on that theme (such as with lowercase letters "eq").
In informal contexts, the notation {''f'' = ''g''} is common.

The definition above used two functions ''f'' and ''g'', but there is no need to restrict to only two functions, or even to only [[finite set|finite]]ly many functions.
In general, if '''F''' is a [[Set (mathematics)|set]] of functions from ''X'' to ''Y'', then the ''equaliser'' of the members of '''F''' is the set of elements ''x'' of ''X'' such that, given any two members ''f'' and ''g'' of '''F''', ''f''(''x'') equals ''g''(''x'') in ''Y''.
Symbolically:
: <math> \operatorname{Eq}(\mathcal{F}) := \{x \in X \mid \forall f,g \in \mathcal{F}, \; f(x) = g(x)\}. </math>
This equaliser may be written as Eq(''f'', ''g'', ''h'', ...) if <math> \mathcal{F}</math> is the set {''f'', ''g'', ''h'', ...}.
In the latter case, one may also find {''f'' = ''g'' = ''h'' = ···} in informal contexts.

As a [[degenerate (math)|degenerate]] case of the general definition, let '''F''' be a [[singleton (set theory)|singleton]] {''f''}.
Since ''f''(''x'') always equals itself, the equaliser must be the entire domain ''X''.
As an even more degenerate case, let '''F''' be the [[empty set]]. Then the equaliser is again the entire domain ''X'', since the [[universal quantification]] in the definition is [[vacuously true]].