Equaliser: Encyclopedia II - Equaliser - Definitions

Equaliser - Definitions

Let X and Y be sets. Let f and g be functions, 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:

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 finitely many functions. In general, if F is a 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:

This equaliser may be denoted Eq(F), or Eq(f,g,h,...) if F 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 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.

Adapted from the Wikipedia article "Definitions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki