Laws
https://web2.0calc.com/formulary/math/algebra/basic-identities
Axioms
A and B
A∪B∅={x:x∈X} Complement: A∪B
A∪(B∪C)=(A∪B)∪C, A∩(B∩C)=(A∩B)∩C A∪(B∩C)=(A∪B)∩(A∪C), A∩(B∪C)=(A∩B)∪(A∩C)
A={a,b,c,d} B={c,d,e,f} A ∪ B = {a,b,c,d,e,f} A ∩ B = {c,d} A - B = {a,b} ≠ B - A = {e,f} C(A)={}
op = ∪, ∩ set operartion
Commutativity: A∪B≡B∪A
Associativity, e.g. (A∪B)∪C≡A∪(B∪C)
Distributivity, e.g. A∪(B∩C)≡(A∪B)∩(A∪C)
Idempotency: ∀A:A∪A≡A,A∩A≡A
Identity: ∀A: A∪∅=A,A∩∅=∅,A∪U=U,A∩U=A
Transitivity: (A≤B≤C)→(A≤C)
Involution: ∀A:A≡¬¬A
De Morgan's Law: supports in proving tautologies and contradiction.
Maths › Algebra › Basic Identities
Basic Identities
∀a,b,c,x∈R
Additive Identity: a+ϵ=a
Additive Inverse: a+(−a)=ϵ=0
Associativity: (a+b)+c=a+(b+c)
Commutativity: a+b=b+a
Subtraction: a−b=a+(−b)
Multiplicative Identity: a∗1=a
Multiplicative Inverse: a∗(1/a)=1(ifaisnot0)
Multiplication times 0:a∗0=0
Associative of Multiplication: (a∗b)∗c=a∗(b∗c)
Commutative of Multiplication: a∗b=b∗a
Distributivity: a(b+c)=ab+ac
Division: ba=ab1
Exponents
xa∗xb=xa+bxa/xb=xa−bax bx=abxax/bx=(a/b)x(xa)b=xab=(xb)a2k=2k−1+2k−1=2⋅2k−1xa/b=bxa3x5 Logarithms Definition:x=logb(a)⟺bx=a(ifa,b>0andb=1) Laws:log(a∗b)=log(a)+log(b)log(ab)=b∗log(a)log(a/b)=log(a)−log(b) Identities
xba=axb
xa/b=axb