This is a task from my exercise sheet in discrete math that lacks a hindsight.
Put A={1,2,3} and form the relation R on A by putting R={(1,1),(2,2),(1,2),(2,3),(3,1)}.
Investigate if R is antisymmetric.
My attempt:
Anti-symmetry means xRy ^ yRx => x=y. Since we don't have (2,1) (3,2) or (1,3) (which would mean x!=y) in R, but we find (1,1) and (2,2) in R, we draw the conclusion that xRy and yRx only when x=y, which is the condition for anti-symmetry? Would be happy if someone can explain this in a different way. Thank you,
Best Answer
$xRy$ and $yRx \implies x=y$ means that if $x\ne y$ we never have both $xRy$ and $yRx$.
So check each pair:
$\{1,2\}$: We have $(1,2)$ so $1R2$ but not $(2,1)$ so $1\not R2$ so we don't have both.
$\{1,3\}$: We have $(3,1)$ so $3R1$ but not $(1,3)$ so $1\not R3$ so we don't have both.
$\{2,3\}$: We have $(2,3)$ so $2R3$ but not $(2,3)$ so $3\not R2$ so we don't have both.
So it is anti-symmetric.
Finding cases where $xRx$ aren't relevant. If $x=y$ then $xRy$ and $yRx$ are just two ways of saying $xRx$ and if you have $\color{blue}xR\color{red}x$ of course you also have $\color{red}xR\color{blue}x$ because they are the same statement and can't help being "trivially self-symmetric" and are not considered to be an issue to $R$ being symmetric or anti-symmetric.
Another way one can define antisymmetric could be:
Definition 2: For any pair $x\ne y$ it is never the case that both $xRy$ ad $yRx$ is true.
Or Definition 3: For any pair $x\ne y$ then: Whenever we have $xRy$ we do not have $yR x$.
Also note: you don't need either $x Ry$ nor $yRx$. You can have one, or the other, or neither; you just can't have both.
So if $Q= \{(1,1),(2,2),(3,3)\}$ then $Q$ is (vacuously) anti-symmetric as well.