I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?
Do you realize why you are having difficulty with this?
You have absolutely no idea what any of this is about.
Given any non-empty set $\displaystyle A$ if $\displaystyle \mathcal{R}\subseteq(A\times A)$
then $\displaystyle \mathcal{R}$ is a relation on $\displaystyle A$.
The diagonal is $\displaystyle \Delta_A=\{(x,x):~x\in A\}$.
If $\displaystyle \Delta_A\subseteq\mathcal{R}$ then $\displaystyle \mathcal{R}$ is reflexive PERIOD!
Go back to reply #2. Does that relation contain the diagonal?
If it does then the relation is reflexive regardless of what else it contains.
If $\displaystyle \mathcal{R}=\mathcal{R}^{-1}$ then it is symmetric regardless. PERIOD!
The following is loosely copied from: http://www.cs.pitt.edu/~milos/course...es/Class21.pdf
Def: R on S1 S2 is a binary association between elements of S1 and S2.
Ex: S1={1,2,3}, S2={A,B,C}. R=(1,A) (2,3) (2,C)
Def: R on S1 is a binary association between elements of S1.
Ex: S1={1,2,3}. R=(1,2) ( 1,1) (3,3)
Def: R on A is reflexive if (a,a) ε R for every element of A.
Def: R on A is symmetric if (a,b) ε R → (b,a) ε R
Def: R on A is antisymmetric if [(a,b) ε R and (b,a) ε R] → a=b
If R is reflexive it might not be antisymmetric. (a,b) and (b,a) may belong to R but a unequal b
The possible relations on {1.2.3} are:
{(1,1), (2,2), (3,3)}
{(1,2), (2,3), (1,3)}
{(2,1) (3,2), (3,1)}
{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}
{(1,1), (2,2), (3,3), (2,1), (3,2), (3,1)}
Unless you consider 1,2,3 to be symbols.
Sure, I can DEFINE things like 1=2, but they are meaningless for the integers.