Equivalence class

Main article: Equivalence class

The set of all a and b for which a ~ b holds make up an equivalence class of X by ~. Let denote the equivalence class to which a belongs. Then all elements of X equivalent to each other are also elements of the same equivalence class.

13. Real numbers possess an ordering relation. This relation we denote by the symbol “” which is read as “greater than”. The axioms of order in based on “” are:

If, then one and only one of the following in true .

If and , then .

If and , then .

If and , then .

The following axioms make clear the notion of a point lying between two other points.

1. When B is between A and C then, A, B and C are distinct points lying on a line and B is between C and A.

2. Given a pair of points A and B there is a point C so that B is between A and C.

3. If B lies between A and C then A does not lie between B and C.

4. Let A, B and C be three points on a plane and a be a line on that does not contain any one of these points. If there is a point D on a that is between A and B then either a contains a point between A and C or a contains a point between B and C.

14. A linear order (also called pseudo-order, according to Wikipedia) is the irreflexive version of a total order. A linearly ordered set, or loset, is a set equipped with a linear order.

In classical mathematics, the distinction between linear orders and total orders is merely a terminological technicality, which is not always observed; more precisely, there is a natural bijection between the set of total orders on a given set S and the set of linear orders on S, and one distinguishes them by the notation < (for the linear order) and ≤ (for the total order). In constructive mathematics, however, they are irreducibly different.