Proofs Flashcards
ab and a+b unique real number
Axiom of closure
a+b=b+a ab=ba
Commutative Axiom
(a+b)+c=a+(b+c) (ab)c=a(bc)
Associative Axiom
Axioms of Equality
Reflective Axiom
Symmetric Axiom
Transitive Axiom
a=a
Reflective Axiom
If a=b, the b=a
Symmetric Axiom
If a=b and b=c then a=c
Transitive Axiom
a(b+c)=ab+ac
Distributive Axiom
-(-a)=a
Cancellation Property of Opposites
a+0=a 0+a=a
Identity Axiom for Addition
a+(-a)=0 -a+a=0
Axiom of Additive Inverses
-(a+b)=-a+(-b)
Property of the Opposite of a Sum
a-b=a+(-b)
Definition of Subtraction
a•1=a 1•a=a
Identity Axiom for Multiplication
a•0=0 0•a=0
Multiplicative Property of Zero
a(-1)=-a (-1)a=-a
Multiplicative Property of -1
-a(b)=-ab a(-b)=-ab (-a)(-b)=ab
Property in Opposite Products
a•1/a=1 1/a•a=1
Axiom of Multiplicative Inverses
1/ab=1/a•1/b If a≠0, b≠0
Property of the Reciprocal of a Product
a÷b=a•1/b If b≠0
Definition of Divsion
Some smaller things
Don’t Distribute the negative, use prop. Opp. Sum.
You can use opposite of any property
Change subtraction to addition (def . Of sub)
Change division to multiplication (def. Of mult)
Distribute and then Prop. Opp. Product
