TRANSITIVE RELATION Meaning and
Definition
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A transitive relation is a term used in mathematics to describe a relation between two or more elements of a set. More specifically, it refers to a relation where if element A is related to element B, and element B is related to element C, then element A is also related to element C. In other words, it is a relation that exhibits the property of transitivity.
For instance, consider a set of people in which the relation represents the concept of "is older than." If person A is older than person B, and person B is older than person C, then it follows that person A is also older than person C. This illustrates the transitive property of the relation.
In formal terms, a relation R on a set A is transitive if for any elements x, y, and z in A, whenever x is related to y and y is related to z, then x is also related to z. Mathematically, it can be expressed as (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R.
Transitive relations are commonly used in various mathematical fields, such as graph theory, where they are useful to model and analyze relationships between objects or entities. Understanding and identifying transitive relations play a fundamental role in various mathematical proofs and reasoning.
Etymology of TRANSITIVE RELATION
The word "transitive" comes from the Latin word "transitus", which means "to go across" or "to pass over". In mathematics, a relation is said to be transitive if, for any three elements a, b, and c, if a is related to b and b is related to c, then a is also related to c. The term "transitive relation" itself was likely coined in the early development of set theory and logic in the 19th and 20th centuries, as mathematicians explored various properties and types of relations.
