fns1 http://www.openmath.org/cd/fns1.ocd 2003-04-01 2001-03-12 2 0 official quant1 relation1 logic1 set1 This CD is intended to be `compatible' with the corresponding elements in Content MathML. In this CD we give a set of functions concerning functions themselves. Functions can be constructed from expressions via a lambda expression. Also there are basic function functions like compose, etc. domainofapplication The domainofapplication element denotes the domain over which a given function is being applied. It is intended in MathML to be a more general alternative to specification of this domain using such quantifier elements as bvar, lowlimit or condition. domain This symbol denotes the domain of a given function, which is the set of values it is defined over. x is in the domain of f if and only if there exists a y in the range of f and f(x) = y range This symbol denotes the range of a function, that is a set that the function will map to. The single argument should be the function whos range is being queried. It should be noted that this is not necessarily equal to the image, it is merely required to contain the image. the range of f is a subset of the image of f image This symbol denotes the image of a given function, which is the set of values the domain of the given function maps to. x in the domain of f implies f(x) is in the image f identity The identity function, it takes one argument and returns the same value. for all x | identity(x)=x left_inverse This symbol is used to describe the left inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the left inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this left inverse. right_inverse This symbol is used to describe the right inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the right inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this right inverse. inverse This symbol is used to describe the inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the inverse function is ill-defined without further stipulations. No assumptions are made on the semantics of this inverse. (inverse(f))(f(x)) = x if f is injective, that is (for all x,y | f(x) = f(y) implies x=y) implies (inverse(f))(f(z)) = z f(inverse(f(y))=y if f is defined at inverse(f)(y) that is, if there exists an x s.t. inverse(f)(y) = x then this implies f(inverse(f)(y)) = y left_compose This symbol represents the function which forms the left-composition of its two (function) arguments. for all f,g,x | left_compose(f,g)(x) = f(g(x)) lambda This symbol is used to represent anonymous functions as lambda expansions. It is used in a binder that takes two further arguments, the first of which is a list of variables, and the second of which is an expression, and it forms the function which is the lambda extraction of the expression An example to show the connection between curried and uncurried applications of a binary function f (lambda(x,y).(f))(a,b)= (lambda(x).((lambda(y).(f))(b)))(a)