TITLE: Example of a relation that is reflexive but not symmetric
QUESTION [8 upvotes]: By definition, $R$, a relation in a set X, is reflexive if and only if $\forall x\in X$, $x\,R\,x$, and $R$ is symmetric if and only if $x\,R\,y\implies y\,R\,x$. 
I think $x\,R\,x$ can also be symmetric when I read the definition, but I also feel there's something wrong or missing in my understanding. 
Can you give an example of a relation that is reflexive but not symmetric?

REPLY [4 votes]: Somewhere, there's a list that shows relations can be any combination of reflexive, symmetric and transitive (despite the famous false proof that symmetric + transitive -> reflexive).  – barrycarter 3 hours ago

Well, I couldn't find one to link to in a few minutes, so let me provide one here.
On the three-element set $\{a, b, c\}$, the following relations are:

Transitive, symmetric: $R_0 = \emptyset$
Transitive, not symmetric: $R_1 = \{(a,b)\}$
Not transitive, not symmetric: $R_2 = \{(a,b), (b,c)\}$
Not transitive, symmetric: $R_3 = \{(a,b), (b,a), (b,c), (c,b)\}$

None of the relations above are reflexive, but they can all be turned into reflexive relations, without affecting their transitivity or symmetry, by adding $R^* = \{(a,a), (b,b), (c,c)\}$ to them.
(In particular, $R_1 \cup R^* = \{(a,a), (a,b), (b,b), (c,c)\}$ is a reflexive, non-symmetric relation on the set $\{a, b, c\}$.  Of course, the restriction of this relation to the two-element subset $\{a, b\}$ yields an even simpler example.)