contradiction
a statement that is always false
biconditional operation
p <-> q; p if and only if q
conditional law
p -> q = notp or q; p <-> q = (p -> q) and (q ->p); p <-> q = (p and q) or (notp and notq)
domination laws
p and false = false; p or true = true
absorption law
p and (p or q) = p
De Morgan’s law
not(p and q) = notp or notq
double negation law
not(not(p)) = p
commutative law
p and q = q and p; p or q = q or p
associative law
p and (q and r) = (p and q) and r; applies if sign is the same
idempotent law
p and p = p; p or p = p
identity laws
p and true = p; p or false = p; situations in which p overrides raw truth values
complement laws
p and notp = false; p or notp = true
distributive laws
p and (q or r) = (p and q) or (p and r); p or (q and r) = (p or q) and (p or r)
idempotent laws
A union A = A;
A intersection A = A;
associative laws
if signs/relations are the same, parantheses can be moved to include or exclude other terms
commutative laws
union and intersection are not directional; A and B can be switched
distributive laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
identity laws
A or null = A;
A and universal set = A;
domination laws
A and null = null;
A or universal set = universal set
double complement law
a complement of a complement is the original
complement laws
A and the complement of A = nothing;
A or the compliment of A = everything / Universal set;
compliment of universal set is nothing, compliment of null is universal set
De Morgan’s laws
compliment of (A or B) = compliment of A and compliment of B;
et vice
Absorption laws
A or (A and B) = A;
A and (A or B) = A
