Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Equaliser (mathematics)
Summary
In mathematics, an equaliser is a set of arguments where two or more functions have equal values.
An equaliser is the solution set of an equation.
In certain contexts, a difference kernel is the equaliser of exactly two functions.
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 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 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.
A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f, g), Ker(f, g), or Ker(f − g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f − g. Furthermore, the kernel of a single function f can be reconstructed as the difference kernel Eq(f, 0), where 0 is the constant function with value zero.
Of course, all of this presumes an algebraic context where the kernel of a function is the of zero under that function; that is not true in all situations.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.