Laws
Axioms
A and B
AβͺBβ
={x:xξ βX} Complement: AβͺB
Aβͺ(BβͺC)=(AβͺB)βͺC, Aβ©(Bβ©C)=(Aβ©B)β©C Aβͺ(Bβ©C)=(AβͺB)β©(AβͺC), Aβ©(BβͺC)=(Aβ©B)βͺ(Aβ©C)
A={a,b,c,d} B={c,d,e,f} A βͺ B = {a,b,c,d,e,f} A β© B = {c,d} A - B = {a,b} β B - A = {e,f} C(A)={}
op = βͺ, β© set operartion
Commutativity: AβͺBβ‘BβͺA
Associativity, e.g. (AβͺB)βͺCβ‘Aβͺ(BβͺC)
Distributivity, e.g. Aβͺ(Bβ©C)β‘(AβͺB)β©(AβͺC)
Idempotency: βA:AβͺAβ‘A,Aβ©Aβ‘A
Identity: βA:Β Aβͺβ
=A,Aβ©β
=β
,AβͺU=U,Aβ©U=A
Transitivity: (Aβ€Bβ€C)β(Aβ€C)
Involution: βA:A⑬¬A
De Morgan's Law: supports in proving tautologies and contradiction.
Maths βΊ Algebra βΊ Basic Identities
Basic Identities
βa,b,c,xβR
Additive Identity: a+Ο΅=a
Additive Inverse: a+(βa)=Ο΅=0
Associativity: (a+b)+c=a+(b+c)
Commutativity: a+b=b+a
Subtraction: aβb=a+(βb)
Multiplicative Identity: aβ1=a
Multiplicative Inverse: aβ(1/a)=1(ifaisnot0)
Multiplication times 0:aβ0=0
Associative of Multiplication: (aβb)βc=aβ(bβc)
Commutative of Multiplication: aβb=bβa
Distributivity: a(b+c)=ab+ac
Division: baβ=ab1β
Exponents
xaβxb=xa+bxa/xb=xaβbaxΒ bx=abxax/bx=(a/b)x(xa)b=xab=(xb)a2k=2kβ1+2kβ1=2β
2kβ1xa/b=bxβa3xβ5Β LogarithmsΒ Definition:x=logb(a)βΊbx=a(ifa,b>0andbξ =1)Β Laws:log(aβb)=log(a)+log(b)log(ab)=bβlog(a)log(a/b)=log(a)βlog(b) Identities
xbaβ=axβb
xa/b=axβb