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.
Transitive relations are quite common. However, one shoud be careful not to grant transitivity when it is not present.
For ecxample, the contextual relation "is preferred to" is not transitive.
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.
are next-door neighbours is not but is neighbour of is!
Such contextual relations can be referred to as "transitive preference" and "intransitive preference."