Transitive Relations

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Mathematical Definition

A binary relation R defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.


Examples of Transitive Relations

R Explanation
precedes if A precedes B, and B precedes C, necessarily A precedes C. This would be true whatever the nature of A, B, and C,although the ideas can be made more precise by thinkingof A, B, and C as events occurring at specific times.
includes if A includes B, and B includes C, necessarily A includes
included in
is less than
is greater than
supports
implies
causes

The generic phrase 'is subordinate to' can be used to represent any of the above relations.

Without it, the development of knowledge would have been considerably more difficult. On the other band, its prevalence may induce a tendency to take it for granted when it is not present. The contextual relation "is preferred to" is a good case in point. Preference is subjective, and subjective relations may or may not be transitive. Thus if a person says "blue is preferred to red" and "red is preferred to yellow," it still may be that the person will say "yellow is preferred to blue," hence, transitivity is violated. For this particular contextual relation, one can speak of"transitive preference" and "intransitive preference."

This mathematical definition of transitivity can be readily interpreted for many contextual relations.