The inequality relations have the property that $x<y$ iff $y>x$. The relation '>' has an opposite, '<', which works the other way round.

Are there any other pairs of operators which work this way?

@christianp Subset/superset?

@christianp You can always define a dual relation $R'$ for a given relation $R$ so that $x R'y$ iff $yRx$.

@Breakfastisready I'm interested in relations for which conventionals symbols for R and R' already exist

@christianp Maybe this is too obvious, but ≤ and ≥ are just as symmetric.

@11011110 yeah, got those

@christianp Surely someone must have used xЯy ?

@christianp Let R be a relation. Principia Mathematica uses R with a right-pointing arrow above to denote the derived relation consisting of all pairs <x, { y | xRy}>. (That is, it relates x to the /set/ of all y that are R-related to x.) I'll write this as “R→”. Then it uses R with a left-pointing arrow (R←) to denote the converse relation of all pairs <{y | xRy}, x>. So x R→ y if and only if y R← x.
Sadly, PM does not use Я to mean the converse of R. It uses “Cnv R”.

@christianp That last sentence was not quite right. It uses “Cnv‘R”, not “Cnv R”. “Cnv” itself denotes the relation consisting of all paird <R, S> where R is the converse of S. (That is, where aRb if and only if bSA.)

@christianp ⊆ and ⊇ (for subsets), ⊂ and ⊃ (for proper subsets), ∈ (in) and ∋ (contains), ⇒ (implies) and ⇐ (is implied by), ≼ and ≽ (for weak partial orders), ≺ and ≻ (for strict partial orders), ⊴ and ⊵ (for normal subgroups), ⊲ and ⊳ (for proper normal subgroups), and trivially, = and =, ≡ and ≡, ∼ and ∼, and so on.

@jeffgerickson @christianp actually $\sim$ isn't horizontally symmetric.

@olligobber @christianp No, but it is rotationally symmetric!

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