Topology and functions

Topology Sets and Functions (XXXIII) Functions Two mappings f : X → Y and g : X → Y are said to be equal ( and we write this f = g ) if f(x) = g(x) for every x in X. Let f, g and h be any three mappings of a non-empty set X into itself, and show that multiplication of mappings is associative in the sense that f(gh) = (fg)h. See the attached file.

Topology and mapping functions

Topology Sets and Functions (XXXIV) Functions Let X be a non-empty set. The identity mapping ix on X is the mapping of X onto itself defined by ix(x) = x for every x. Thus ix sends each element of X to itself; that is, it leaves fixed each element of X. Show that f ix = ix f = f for any mapping f of X into itself. See the attached file.

Topology questions

Topology Sets and Functions (XXXV) Functions Let X be a non-empty set. The identity mapping ix on X is the mapping of X onto itself defined by ix(x) = x for every x. Thus ix sends each element of X to itself; that is, it leaves fixed each element of X. If f is one-to-one onto, so that its inverse f– 1 exists, show that f f– 1 = f– 1 f = ix . See the attached file.

Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is one-to-one iff there exists a mapping g of Y into X such that gf = iX.

Topology Sets and Functions (XXXVII) Functions Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is one-to-one iff there exists a mapping g of Y into X such that gf = iX. See the attached file.

Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is onto iff there exists a mapping h of Y into X such that fh = iX.

Topology Sets and Functions (XXXVIII) Functions Let X and Y be non-empty sets and f a mapping of X into Y. Show that f is onto iff there exists a mapping h of Y into X such that fh = iX. See the attached file.