Transitive Property

Mathematics is based on certain clearly defined axioms and fundamental relations. These fundamental relations help establish a mathematical theory firmly.

Definition
A binary relation is a bunch of ordered pairs of elements belonging to a set. It is also known by different math terms as a 2-place relation or dyadic relation. Ordered pairs are elements of a set placed in a specific order. For example, the coordinates of a point drawn in a graph form an ordered pair.

A transitive property of a binary relation 'R', defined over a set 'A', is such that if a set element 'a' is related to a set element 'b' and 'b' is further related to 'c' , then 'a' is related to 'c'. Symbolically, it can be defined as follows. For the set of elements a, b and c belonging to a set A, a binary relation '~' has the property defined by,

If a ~ b and b ~ c, then that implies a ~ c.

Let us look at the transitive property, as an essential property of binary relations like equality and inequality.

Examples
Here are some of the examples, applied to the concepts of equality and inequality. One needs to be careful in its application to different binary relations. Every relation may not be transitive.

Transitive Property of Equality
The property of equality is defined in the following way. For three elements a, b and c belonging to set A, the property is defined as:

If a = b and b = c, then a = c

This is a simple enough relation to understand and is just commonsense stated in mathematical language!

Transitive Property of Inequalities
The property for inequalities is defined as follows:

For elements, a, b, c belonging to a set A,

If a > b and b > c, then a > c

If a ≥ b and b ≥ c, then a ≥ c

If a < b and b < c, then a < c

If a ≤ b and b ≤ c, then a ≤ c

There are many more such transitive binary relations that can be defined on a set. However, one cannot go on applying the transitive relation criteria to every binary relation randomly.

For example if Richard is the father of Henry and Henry is the father of John, that doesn't imply that Richard is the father of John. The transitive property depends on the nature of a binary relation. It is one of the essential properties that define a binary relation to be an equivalence relation.

The transitive property plays an important role in ordering of numbers on the real line. It is a simple enough concept to understand but needs to be applied carefully.