Transitive Relations: Difference between revisions
No edit summary |
No edit summary |
||
| Line 9: | Line 9: | ||
=== Examples of Transitive Relations === | === Examples of Transitive Relations === | ||
{| class="wikitable" | {| class="wikitable sortable" style="width: 65%;" | ||
|- | |- | ||
! R | ! scope="col" style="background:#efefef;" align="left"; "width: 25%"| R | ||
! | ! scope="col" style="background:#efefef;" align="left"; "width: 75%"| Explanation | ||
|- | |- | ||
| precedes | | precedes | ||
| Line 39: | Line 39: | ||
The generic phrase ''''is subordinate to'''' can be used to represent any of the above relations. | 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 | This mathematical definition of transitivity can be | ||
Revision as of 14:10, 9 January 2022
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.