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i. {(a, b) : a and b have met} ii. {(a, b)} : a and b speak a common language

i) Reflexive: yes Symmetric: yes Transitive: No, if a met b and b met a then a does not met c.

ii) Reflexive: yes Symmetric: yes Transitive: No, if a speak Spanish with b and b speck French with c then that does not mean a speak French with c.

Please need more simple elaborations and comments.

Surdz
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  • I agree. Perhaps a more clear explanation for ii. a speaks Spanish (only), b speaks Spanish and French, c speaks French only. – Doug M May 19 '16 at 02:29
  • I once met a man who worked closely with Paul Erdős and had met him on numerous occasions. Unfortunately, Erdős had died before I had even made it out of middle school and have never had the opportunity to meet him myself. In the same way, for ($i$) it is possible that the set of people you have some person $a$ who has met person $b$ while at the same time the same person $b$ has met some additional person $c$ (you wrote $a$ again here where you should have written $c$) while person $a$ has not met $c$. – JMoravitz May 19 '16 at 02:31
  • Of course, this all depends on the set of people in question. If the set of people are your classmates in a class which is taught in English and all students are expected to know English and in fact do know English, trivially all students are related via the "speaks a common language" relation. There is also the philosophical question of whether a person can have "met himself" or not. – JMoravitz May 19 '16 at 02:33

1 Answers1

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Correct idea. Presentation needs more work.

i) Reflexive: yes Symmetric: yes Transitive: No, if a met b and b met a then a does not met c.

Rather, say "You can find three people such that $a$ met $b$, and $b$ met $c$, but $a$ has not met $c$."   So the relation "met" is not transitive.

(At least, it is not necessarily transitive, depending on the domain - a small enough set of people might conceivably have all met each other.)

ii) Reflexive: yes Symmetric: yes Transitive: No, if a speak Spanish with b and b speck French with c then that does not mean a speak French with c.

You can find three people such that the first and third are monolingual in different languages and the second bilingual in both.   Then the first and second, and the second and third have a common language, but the first and third do not.   Thus "have a common language" is not (necessarily) a transitive relation.

Graham Kemp
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