| Finite Set |
Article Index for Finite |
Website Links For Finite |
Information AboutFinite Set |
| CATEGORIES ABOUT FINITE SET | |
| discrete mathematics | |
| basic concepts in set theory | |
| mathematical terminology | |
| cardinal numbers | |
|
Equivalently, a set is finite if its Cardinality , i.e. the number of its elements, is a natural number. For instance, the set of Integer s between -15 and 3 (excluding the end points) is finite, since it has 17 elements. The set of all Prime Number s is not finite. Sets that are not finite are called Infinite . A set is called '' Dedekind Finite '' if there exists no bijection between the set and any of its proper Subset s. It is a Theorem (assuming the Axiom Of Choice ) that a set is finite if and only if it is Dedekind finite. ALTERNATIVE DEFINITIONS OF FINITE There are many definitions of "finite", including the following. Notice that these are equivalent, if the axiom of choice is true. # A set is finite iff it can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number. # A set is finite iff it has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. # {Link without Title} A set is finite iff it can be given a total ordering which is both well ordered forwards and backwards. That is, a set is finite iff every non-empty subset has both a least and a greatest element in the subset. # {Link without Title} A set is finite iff every function from the set one-to-one into itself is onto. # A set is finite iff every function from the set onto itself is one-to-one. # {Link without Title} A set is finite iff every non-empty family of subsets of the set has a minimal element wrt inclusion. FOOTNOTES SEE ALSO |
|
|