Order: An order on a set is defined as a relation that is reflexive, transitive and antisymmetric i.e
A function is said to be order preserving if . A set is said to be ordered or simply ordered if for every pair of elements and in M, or . If this is not true for every pair, the set is said to be partially ordered. A mapping is said to be an order isomorphism if .
Two sets that are order isomorphic to each other are said to have the same order type. If a set is finite with elements, it has order type . The order type of the set of natural numbers is . Hence, a set having order type has power . But a set having power need not have order type since the set can be ordered in many ways (in fact, uncountably many ways) and hence can have uncountably many order types.
We say that if and .
Well ordered set: A non empty set S is said to be well ordered if every non empty subset of S has a least element i.e. and a Q such that .
eg: The set of natural numbers is well ordered. The intervals [0,1], (2,100) are not well ordered since not every subset has a least element.