Equaliser (mathematics)
In mathematics, an equalizer is a set of arguments where two or more functions have equal values. An equalizer is the solution set of an equation. In certain contexts, a difference kernel is the equalizer of exactly two functions.
Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equalizer of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically:
The equalizer 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 finitely many functions. In general, if F is a set of functions from X to Y, then the equalizer 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:
This equaliser may be written as Eq(f,g,h,...) if is the set {f,g,h,...}. In the latter case, one may also find {f = g = h = ···} in informal contexts.
As a degenerate case of the general definition, let F be a singleton {f}. Since f(x) always equals itself, the equalizer must be the entire domain X. As an even more degenerate case, let F be the empty set {}. Then the equalizer is again the entire domain X, since the universal quantification in the definition is vacuously true.
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