Proving reflexivity, symmetry and transitivity,..., on a relation on words The relation R ,$uRv$ is defined iif a word u is the suffix of a word v. u is a suffix of v if there exist another word w such that $v = wu $
I have to verify the 6 following relations.


*

*Reflexive : Yes because $wu = wu$

*Symmetric : No because $v = wu$     , $u \ne wu$

*Transitive : Yes , example : u="to", v="potato"   and  u2="otato", v="potato"  and uRu2 

*Asymmetric : No, v = wu, v $\ne$ u , except if w is an empty word 

*Antisymmetric : No. ex: "to" R "potato" but "to" $\ne$ "potato"

*Irreflexive : No. ex: $wu = wu$.


Can you help me for those that are incorrect

Show reflexivity using the empty word. Your proof isn't correct there.
For symmetry, specify that if $w$ is non-empty, then $u\:R\:wu$ but it is not the case that $wu\:R\:u$.
Your example for transitivity is good, but you want a more general argument. If $u\:R\:v$ and $v\:R\:w$, then there exist words $w_1,w_2$ such that $v=w_1u$ and $w=w_2v$. Now, $w_2w_1$ is a word, and $w=w_2w_1u$, so $u\:R\:w$.
A (non-empty) reflexive relation cannot be asymmetric, nor can it be irreflexive. (Why?) This should suggest how to show that $R$ is not asymmetric or irreflexive.
Actually, your relation is antisymmetric. You must show that if $u$ is a suffix of $v$ and $v$ is a suffix of $u,$ then in fact $u$ and $v$ are the same word. Think on it a bit.
