I'm having trouble understanding whether or not relations are reflexive, symmetric and transitive. I know that for a relation to be any of those it must satisfy the conditions:
- reflexive: for every $s \in S$ $sRs$ (s is related to itself and therefore reflexive)
- symmetric: for every $s,t \in S$, if $sRt$ then $tRs$
- transitive: for every $s,t,u \in S$, if $sRt$ and $tRu$ then $sRu$
However I don't quite understand how to apply these conditions to problems. For example how would I solve something like this:
a) $x\sim y$ if $x$ and $y$ are people and there exists a country $C$ such that $x$ has been to country $C$ and $y$ has been to country $C$.
b) $x\sim y$ if $x$ and $y$ are strings which contain a common character.