boolean

we have 1 + 10 useful boolean functions: not, and or xor imp con nand nor nxor nimp ncon

we can neatly enumerate all the possible boolean functions: with n inputs, we have 2ⁿ possible permutations, for which we have 2^2ⁿ functions.

nullary functions

we have 2^2⁰ = 2 nullary "functions":

name	value	symbol	description
`false`	❌	⊥	boolean bottom element
`true`	✅	⊤	boolean top element

these are stored as constants instead

unary functions

with n = 1 inputs, we have 2^2¹ = 4 unary functions:

name	F T	symbol	description
`false`	❌❌	⊥	constant always-false function
`id`	❌✅	id	identity
`not`	✅❌	¬	negation
`true`	✅✅	⊤	constants always-true function

not is the only useful non-degenerate function here

binary functions

with n = 2 inputs, we have 2^2² = 16 binary functions:

name	FF FT TF TT	symbol	description
`false`	❌❌❌❌	⊥	constant always-false function
`and`	❌❌❌✅	∧	conjunction
`nimp`	❌❌✅❌	↛	abjunction
`ncon`	❌✅❌❌	↚	converse abjunction
`nor`	✅❌❌❌	↓	joint denial
`fst`	❌❌✅✅	π₁	first argument
`snd`	❌✅❌✅	π₂	second argument
`xor`	❌✅✅❌	↮	exclusive disjunction
`nxor`	✅❌❌✅	↔	material biconditional
`nsnd`	✅❌✅❌	¬π₂	¬first argument
`nfst`	✅✅❌❌	¬π₁	¬second argument
`or`	❌✅✅✅	∨	disjunction
`con`	✅❌✅✅	←	converse material implication
`imp`	✅✅❌✅	→	material implication
`nand`	✅✅✅❌	↑	alternative denial
`true`	✅✅✅✅	⊤	constant always-true function

fst, snd, nsnd, nfst are degenerate functions

ternary functions and beyond

with n = 3 inputs, we have 2^2³ = 256 ternary functions. since there are so many, and because they can be composed from binary functions anyway, daamath does not maintain functions of n ≥ 3.