Implementation of mathematics in set theory: Difference between revisions

Revision as of 03:22, 21 December 2005

Revision as of 03:33, 21 December 2005

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and further it can be proved that there is no such function: see the resolution of

Cantor's paradox in the [[New Foundations]] article).

=== Operations on functions ===

The composition <math>g \circ f</math> of functions f and g is defined as the relative

product <math>f | g</math>, if this is a function: we have <math>g \circ f</math> a function with <math>(g \circ f)(x) = g(f(x))</math> if the range of f is a subset of the ___domain of g. The inverse <math>f^{-1}</math>

is the converse of f (if this is a function). The identity function <math>i_A</math> is the

set <math>\{(x,x)\mid x \in A\}</math> for any set A: this is a set in both theories for different

reasons.

=== Special kinds of function ===

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Notice that our terminology here adjusts for the fact that functions as we have defined

them do not determine their codomains.

== Size of sets ==

In both [[ZFC]] and [[NFU]], we say that two sets A and B are the same size (or are <strong>equinumerous</strong>) if and only if there is a bijection f from A to B.

== Finite sets and natural numbers ==