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I understand that what a preorder relation is, and I understand what a symmetric preorder –equivalence relation – is, but I don't quite see the logic behind anti-symmetry? For all the examples I've seen where the relation IS anti-symmetric, it has been the case where there was NOT an (x,y) and (y,x) inside the relation, so the following y = x assertion could not be tested.

For example: {(a,a),(b,b),(c,c)} is both symmetric and anti-symmetric because each's "if" conditions fail to hold?

Any help/examples would be appreciated.

tom_croose
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  • Welcome to MSE! Consider the relation "less than or equal to" over $\mathbb R$. If $x \le y$ and $y \le x$, then $x=y$. – Sávio Apr 01 '21 at 02:51
  • Thank you @Sávio, in my example above, what would be the correct reasoning as to why it would be both anti-symmetric and symmetric? Edit: does it just depend on what relationship you are trying to enforce? – tom_croose Apr 01 '21 at 02:53

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Here are four pictures. These give examples of a three-element set and four relations on it. Each relation consists of three ordered pairs. The four examples show all the possible combinations of satisfying and not satisfying the properties of being symmetric and of being anti-symmetric.

pictures of four relations as described in text

These can be described symbolically, assuming the underlying set is $\{a,b,c\}$.

$\{(a,a), (b,b), (c,c)\}$ is both symmetric and anti-symmetric. (top left in image)

$\{(a,a), (b,c), (c,b)\}$ is symmetric and not anti-symmetric. (top right in image)

$\{(a,b), (b,c), (c,a)\}$ is not symmetric and is anti-symmetric. (bottom left in image)

$\{(a,b), (b,c), (c,b)\}$ is not symmetric and is not anti-symmetric. (bottom right in image)

John Stell
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