Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)

Question

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?

Answer 1

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

Answer 2

No, there are plenty of anti-symmetric relations that are not reflexive.

For instance, let $R$ be the relation $R=\{(1,2)\}$ on the set $A=\{1,2,3\}$. This relation is certainly not reflexive, but it is in fact anti-symmetric. This is vacuously true, because there are no $x$ and $y$, such that $(x,y)\in R$ and $(y,x)\in R$.

Edit: Why is this anti-symmetric? Because in order for the relation to be anti-symmetric, it must be true that whenever some pair $(x,y)$ with $x\neq y$ is an element of the relation $R$, then the opposite pair $(y,x)$ cannot also be an element of $R$. This is true for our relation, since we have $(1,2)\in R$, but we don't have $(2,1)$ in $R$.

Also, the relation $R=\{(1,2),(2,3),(1,1),(2,2)\}$ on the same set $A$ is anti-symmetric, but it is not reflexive, because $(3,3)$ is missing.

As for a reflexive relation, which is not anti-symmetric, take $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$.

Answer 3

They're two different things, there isn't really a strong relationship between the two.

Based on the definitions you're using, they both give two different criteria for concluding that $(x, x) \in R$.

For any antisymmetric relation $R$, if we're given two pairs, $(x, y)$ and $(y, x)$ both belonging to $R$, then we can conclude that in fact $x = y$, so that that

$$(x, y) = (x, x) = (y, x),$$

and $(x, x) \in R$. It may really be better stated as saying that

$$\text{ If } x \neq y, \text{ then at most one of $(x, y)$ or $(y, x)$ is in $R$}.$$

That is, it may be a bit misleading to even think about $(x,y)$ and $(y, x)$ as being pairs in $R$, since antisymmetry forces them to in fact be the same pair, $(x, x)$.

Antisymmetric relations may or may not be reflexive. $<$ is antisymmetric and not reflexive, while the relation "$x$ divides $y$" is antisymmetric and reflexive, on the set of positive integers.

A reflexive relation $R$ on a set $A$, on the other hand, tells us that we always have $(x, x) \in R$; everything is related to itself. Reflexive relations may or may not be symmetric, or antisymmetric:

$\leq $ is reflexive and antisymmetric, while $=$ is reflexive and symmetric.