is this an equivalence relation? by Reflexive, Symmetric and Transitive.

Question

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.

Answer 1

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.