binary relations¶

when a binary relation rel(a, b) is defined on a carrier set, we can compose rel(a, b) and rel(b, a) using the boolean functions[boolean] to get many useful functions. we shall use the common partial ordering relation ≤ (less than or equal to) to name these composed functions, although the binary relation can be anything: an equivalence relation, a partial ordering, a pre-order, et cetera.

definition	∥	<	>	=	name	description
[false](rel(a, b), rel(b, a))	❌	❌	❌	❌	--	degenerate always-false function
[and](rel(a, b), rel(b, a))	❌	❌	❌	✅	eq	a is equal to b
[ncon](rel(a, b), rel(b, a))	❌	❌	✅	❌	gt	a is strictly greater than b
[nimp](rel(a, b), rel(b, a))	❌	✅	❌	❌	lt	a is strictly lesser than b
[nor](rel(a, b), rel(b, a))	✅	❌	❌	❌	nc	a is not comparable to b
[snd](rel(a, b), rel(b, a))	❌	❌	✅	✅	ge	a ≥ b. this function is degenerate because it is the same as rel(b, a)
[fst](rel(a, b), rel(b, a))	❌	✅	❌	✅	le	a ≤ b. this function is degenerate because it is the same as rel(a, b)
[xor](rel(a, b), rel(b, a))	❌	✅	✅	❌	so	a is strictly ordered with b
[nxor](rel(a, b), rel(b, a))	✅	❌	❌	✅	ns	a is not strictly ordered with b
[nfst](rel(a, b), rel(b, a))	✅	❌	✅	❌	nb	a is not below b. this function is degenerate because it is the same as not(rel(a, b))
[nsnd](rel(a, b), rel(b, a))	✅	✅	❌	❌	na	a is not above b. this function is degenerate because it is the same as not(rel(b, a))
[or](rel(a, b), rel(b, a))	❌	✅	✅	✅	cp	a is comparable to b
[imp](rel(a, b), rel(b, a))	✅	❌	✅	✅	nl	a is not strictly lesser than b
[con](rel(a, b), rel(b, a))	✅	✅	❌	✅	ng	a is not strictly greater than b
[nand](rel(a, b), rel(b, a))	✅	✅	✅	❌	ne	a is not equal to b
[true](rel(a, b), rel(b, a))	✅	✅	✅	✅	--	degenerate always-true function

you may also notice these are the same degenerates as the boolean functions[boolean]; but though they are degenerate, daamath still includes them because ge(a, b) & le(a, b) may be useful aliases for rel(a, b) & rel(b, a). similarly with nb(a, b) & na(a, b) for not(rel(a, b)) & not(rel(b, a)).

total order¶

when rel(a, b) is a total order, cp is always true; so the 16 functions are reduced down to 8. we may present these visually using the one-dimensional nature of a totally ordered set. we have three partitions of the carrier set:

x < A	x = A	x > A	general function	toset equivalent
❌	❌	❌	nc	[false]
❌	❌	✅	nb	gt
❌	✅	❌	ns	eq
✅	❌	❌	na	lt
❌	✅	✅	nl	ge
✅	❌	✅	ne	so
✅	✅	❌	ng	le
✅	✅	✅	cp	[true]

w.r.t. two elements¶

with respect to two elements A and B, we have five partitions of the carrier set:

x < A	x = A	A < x < B	x = B	x > B	name	description
❌	❌	❌	❌	❌	~~false~~
❌	❌	❌	❌	✅	~~gt_B~~
❌	❌	❌	✅	❌	~~eq_B~~
❌	❌	✅	❌	❌	oo	in open-open interval (A, B)
❌	✅	❌	❌	❌	~~eq_A~~
✅	❌	❌	❌	❌	~~lt_a~~
❌	❌	❌	✅	✅	~~ge_B~~
❌	❌	✅	❌	✅	?
❌	❌	✅	✅	❌	oc	in open-closed interval (A, B]
❌	✅	❌	❌	✅	?
❌	✅	❌	✅	❌	?
❌	✅	✅	❌	❌	co	in closed-open interval [A, B)
✅	❌	❌	❌	✅	ncc	not(cc)
✅	❌	❌	✅	❌	?
✅	❌	✅	❌	❌	?
✅	✅	❌	❌	❌	~~le_A~~
❌	❌	✅	✅	✅	~~gt_A~~
❌	✅	❌	✅	✅	?
❌	✅	✅	❌	✅	?
❌	✅	✅	✅	❌	cc	in closed-closed interval [A, B]
✅	❌	❌	✅	✅	nco	not(co)
✅	❌	✅	❌	✅	?
✅	❌	✅	✅	❌	?
✅	✅	❌	❌	✅	noc	not(oc)
✅	✅	❌	✅	❌	?
✅	✅	✅	❌	❌	~~lt_B~~
❌	✅	✅	✅	✅	~~ge_A~~
✅	❌	✅	✅	✅	~~so_A~~
✅	✅	❌	✅	✅	noo	not(oo)
✅	✅	✅	❌	✅	~~so_B~~
✅	✅	✅	✅	❌	~~le_B~~
✅	✅	✅	✅	✅	~~true~~

FAQ¶

"where is the binary relation for these functions stored?" in dm.context — both the carrier set and the binary relation for these functions are stored in dm.context