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.
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 $x$?