TITLE: Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
QUESTION [9 upvotes]: Antisymmetric: $\forall x\forall y[ ((x,y)\in R\land (y, x) \in R) \to x= y]$
reflexive: $\forall x[x∈A\to (x, x)\in R]$
What really is the difference between the two? Wouldn't all antisymmetric relations also be reflexive?

REPLY [8 votes]: Here are a few relations on subsets of $\Bbb R$, represented as subsets of $\Bbb R^2$. The dotted line represents $\{(x,y)\in\Bbb R^2\mid y = x\}$.
Symmetric, reflexive: 
Symmetric, not reflexive 
Antisymmetric, not reflexive 
Neither antisymmetric, nor symmetric, but reflexive 
Neither antisymmetric, nor symmetric, nor reflexive