$R=\{(a,a),(b,c),(c,b)\}$ why is it not antisymmetric

Question

$R=\{(a,a),(b,c),(c,b)\}$ why is it not antisymmetric. I know the definition of antisymmetric

if $(x,y)∈R$ and $(y,x)∈R$, then $x = y$ is true vacuously.

but what does it mean, From my understanding, $(x,y)$ and $(y,x)→x=y$, so $(b,c),(c,b)$ should satisfy it, but why.

I have another relation

$R=\{(a,a),(b,b),(c,c),(a,c)\}$ why is it antisymmetric. appreciate any help

Answer 1

Why do you say "if (x,y)∈R and (y,x)∈R, then x = y is true vacuously.?" The definition of antisymmetric does not include "vacuously". The definition is for all $x,y$. In your example, we have $(b,c) \in R$ and $(c,b) \in R$, but $b \neq c$ so the relation is not antisymmetric.

Answer 2

If it was anti-symmetric, you should have that b=c, but the fact that both $(b,c)$ and $(c,b)$ are listed as elements in the relation tells us that $b \not = c$. So that means the first relation is not anti-symmetric.

Your second relation is anti-symmetric, however, since there you do not have a case where $(x,y)$ and $(y,x)$ for some $x \not = y$

Answer 3

According to definition of antisymmetric relation:

if $(x,y)∈R$ and $(y,x)∈R$, then $x = y$ is true vacuously.

In given relation $R=\{(a,a),(b,c),(c,b)\} $, when $x=b$ and $y=c$ :

if $(b,c)∈R$ and $(c,b)∈R$, then $b = c$ is not true. Since $b\neq c$.

But, when $x=a$ and $y=a$ :

if $(a,a)∈R$ and $(a,a)∈R$, then $a = a$ is true.

Note that in an antisymmetric relation only diagonal element can be present and relation should be asymmetric.

In your second relation $R=\{(a,a),(b,b),(c,c),(a,c)\}$ diagonal element can be present that are $\{(a,a),(b,b),(c,c)\}$ and remaining relation ${(a,c)}$ is asymmetric, so realtion is antisymmetric.

Answer 4

Try to depict definitions about relations as follows (well, at least it helps me): an element $(x,y)$ in your relation is an arrow going from $x$ to $y$. It is a directed arrow!

So the definition of 'antisymmetric' tells you that if you have an arrow from $x$ to $y$ and an arrow from $y$ to $x$, that arrow is actually a loop from $x$ to itself (or $y$ to itself, whatever label you want to give the point).

In your first example, you have an arrow from $b$ to $c$ and one from $c$ to $b$, but it is not a loop since $b$ and $c$ are different points.

In your second example: you have three loops: one on $a$, one on $b$ and one on $c$. Moreover, you have an arrow from $a$ to $c$ (but there is no arrow from $c$ to $a$, so there is no problem for the definition of antisymmetry to hold).