Is there a relation that is neither reflexive nor transitive nor symmetric or anti-symmetric Can anyone help me with this question i got from discrete, perhaps my understanding is not good enough and I've been stuck trying to think of a possible relation that could work.
It asked for a relation on R {a,b,c}
 A: hardmath’s comment is right on the money: you need to look at the definitions of reflexivity, antisymmetry, transitivity, and symmetry and see exactly what it takes to ensure that $R$ does not have each of these properties. Take them one at a time. It helps to have a diagram of $\{a,b,c\}\times\{a,b,c\}$ in front of you, since $R$ is a subset of that set: you can use it to help you keep track of which of the $9$ possible ordered pairs you want to keep in $R$, and which ones you want to keep out of $R$. I’ll give you a good start on the task and leave the rest to you to finish.
$$\begin{array}{ccc}
\langle a,c\rangle&\langle b,c\rangle&\langle c,c\rangle\\
\langle a,b\rangle&\langle b,b\rangle&\langle c,b\rangle\\
\langle a,a\rangle&\langle b,a\rangle&\langle c,a\rangle
\end{array}$$

*

*If $R$ is not reflexive, there must be some $x\in\{a,b,c\}$ such that $x\not\mathrel{R}x$. You can pick any of the members of $\{a,b,c\}$ to fill this rôle; let’s pick $a$ and color $\langle a,a\rangle$ blue to indicate that it definitely is not in $R$:

$$\begin{array}{ccc}
\langle a,c\rangle&\langle b,c\rangle&\langle c,c\rangle\\
\langle a,b\rangle&\langle b,b\rangle&\langle c,b\rangle\\
\color{blue}{\langle a,a\rangle}&\langle b,a\rangle&\langle c,a\rangle
\end{array}$$

*

*If $R$ is not antisymmetric, there must be some $x,y\in\{a,b,c\}$ such that $x\mathrel{R}y$, and $y\mathrel{R}x$, but $x\ne y$. Let’s try using $a$ and $b$; if that doesn’t work out, we can always back up and try a different pair. Using red to indicate ordered pairs that definitely are in $R$, we now have:

$$\begin{array}{ccc}
\langle a,c\rangle&\langle b,c\rangle&\langle c,c\rangle\\
\color{red}{\langle a,b\rangle}&\langle b,b\rangle&\langle c,b\rangle\\
\color{blue}{\langle a,a\rangle}&\color{red}{\langle b,a\rangle}&\langle c,a\rangle
\end{array}$$

*

*If $R$ is not transitive, there must be some $x,y,x\in\{a,b,c\}$ such that $x\mathrel{R}y$, and $y\mathrel{R}z$, but $x\not\mathrel{R}z$. Note that $x,y$, and $z$ are not all required to be distinct. We already have such an example just in the ordered pairs whose status vis à vis $R$ has been decided; I leave it to you to find it.


*If $R$ is not symmetric, there must be some $x,y\in\{a,b,c\}$ such that $x\mathrel{R}y$, but $y\not\mathrel{R}x$. I’ll leave this one to you; there are several ways to complete the construction of $R$ so that it is not symmetric.
