[Math] Proving reflexivity, symmetry and transitivity,…, on a relation on words

Question

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.

1. Reflexive : Yes because $wu = wu$
2. Symmetric : No because $v = wu$ , $u \ne wu$
3. Transitive : Yes , example : u="to", v="potato" and u2="otato", v="potato" and uRu2
4. Asymmetric : No, v = wu, v $\ne$ u , except if w is an empty word
5. Antisymmetric : No. ex: "to" R "potato" but "to" $\ne$ "potato"
6. Irreflexive : No. ex: $wu = wu$.

Can you help me for those that are incorrect

Answer 1

Here a few comments/corrections. (I am assuming that the empty word $\lambda$ is included, otherwise some answers are different.)

1. Reflexive: $\newcommand{\R}{\operatorname{R}} u\R u$ for all $u$. The relation is reflexive because any word is a suffix of itself. (Letting $\lambda$ denote the empty string, $w = \lambda w$.)
2. Symmetric: The relation is not symmetric, which you ought to demonstrate with a counterexample. For example, with $u =$ "at" and $v =$ "cat", $u \R v$. However, $\newcommand{\notR}{\!\not{\!\!\R}} v \notR u$ as "cat" is not a suffix of "at".
3. Transitive: You need to show that if $u \R v$ and $v \R w$ then $u \R w$ for all $u$, $v$, and $w$. Unpackage the definition: $v = xu$ for some $x$ and $w = yv$ for some $y$. Now, $w = y(xu) = (yx)u$, showing that $u \R w$.
4. Asymmetric: The relation is not asymmetric, since $u \R u$.
5. Antisymmetric: You need to show that if $u \R v$ and $v \R u$ then $u = v$. Using the definition of the relation, $v = xu$ for some $x$ and $u = yv$ for some $y$. Thus, $v = x(yv) = (xy)v$, so $xy = \lambda$, which implies that $x = y = \lambda$. (There's no cancellation possible in concatenating strings.)
6. Irreflexive: The relation is reflexive, so it cannot be irreflexive.