alt Hacker NewsUnless there's some idiosyncratic meaning for the `=>`, the Antisymmetry one basically says `Orange -> Yellow => Yellow -/> Orange`. The diagram is not acurate. The prose is very imprecise. "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me." NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
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It really isn't a long enough section to get lost in.
The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.
> NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.
I like the article's "imprecise prose" better:
You have x ≤ y and y ≤ x only if x = y
I don't think they are completely wrong - "=>" is just implication. A hidden assumption in their diagrams is that circles of different colours are assumed to be different elements.
A morphism from orange to yellow means "O <= Y". From this, antisymmetry (and the hidden assumption) implies that "Y not <= O".
Totality is just the other way around (all two distinct elements are comparable in one direction).