Trouble determining whether relations are reflexive, symmetric and transitive. 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.
 A: Perhaps it will help if you literally write out what each statement means in each case. 
Let's try another example then you can do your examples. Let our overlying set be the set of all animals.
$x \sim y$ if $x$ is from the same species as $y$.
1) Reflexive: We are asking if $x$ is from the same species as $x$.
2) Symmetric: Assuming $x$ is from the same species as $y$. Is $y$ from the same species as $x$? 
3) Transitive: Assuming $x$ and $y$ are from the same species, and $y$ and $z$ are from the same species. Are $x$ and $z$ from the same species? 
Lets try to answer property three by thinking intuitively.
We assume $x$ and $y$ are of the same species. Without loss of generality, lets say that species is tigers. So we know that $x$ and $y$ are both tigers. 
Then we also assume that $y$ and $z$ are of the same species. But we know $y$ is a tiger, so what species is $z$? is this the same species as $x$?
A: Maybe you can consider the relation $\sim$ to be friendship and that would help? Let $S$ be the collection of all people living in your city.
Reflexive: Is everyone their own friend?
Symmetry: If Bob is a friend Alice, is Alice necessarily a friend of Bob?
Transitive: If Kyle is a friend of Judith, and Judith is a friend of Joe, is Kyle necessarily a friend of Joe?
