relational

we have 14 useful relational functions: ev gt lt ic ge le so nso nle nge nic nlt ngt nev

when a binary relation rel(a, b) is defined on a carrier set, we can combine rel(a, b) and rel(b, a) using the binary boolean functions to get many useful functions. to name these combined functions, we'll use the common partial ordering relation ≤ (less than or equal to), but note that the binary relation can be anything: an equivalence relation, a partial ordering, a pre-order, et cetera.

name	combiner	∦ < > ∼	symbol	description
`false`	`false`	❌❌❌❌	⊥	constant always-false function
`ev`	`and`	❌❌❌✅	∼	equivalent
`gt`	`ncon`	❌❌✅❌	>	greater than
`lt`	`nimp`	❌✅❌❌	<	lesser than
`ic`	`nor`	✅❌❌❌	∦	incomparable
`ge`	`snd`	❌❌✅✅	≥	greater than or equivalent
`le`	`fst`	❌✅❌✅	≤	lesser than or equivalent
`so`	`xor`	❌✅✅❌	≶	strictly ordered
`nso`	`nxor`	✅❌❌✅	≸	not strictly ordered
`nle`	`nfst`	✅❌✅❌	≰	not lesser than nor equivalent
`nge`	`nsnd`	✅✅❌❌	≱	not greater than nor equivalent
`nic`	`or`	❌✅✅✅	∥	not incomparable
`nlt`	`imp`	✅❌✅✅	≮	not lesser than
`ngt`	`con`	✅✅❌✅	≯	not greater than
`nev`	`nand`	✅✅✅❌	≁	not equivalent
`true`	`true`	✅✅✅✅	⊤	constant always-true function

relational

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