Proofs

Question 1

ab and a+b unique real number

Answer 1

Axiom of closure

Question 2

a+b=b+a ab=ba

Answer 2

Commutative Axiom

Question 3

(a+b)+c=a+(b+c) (ab)c=a(bc)

Answer 3

Associative Axiom

Question 4

Axioms of Equality

Answer 4

Reflective Axiom
Symmetric Axiom
Transitive Axiom

Question 5

a=a

Answer 5

Reflective Axiom

Question 6

If a=b, the b=a

Answer 6

Symmetric Axiom

Question 7

If a=b and b=c then a=c

Answer 7

Transitive Axiom

Question 8

a(b+c)=ab+ac

Answer 8

Distributive Axiom

Question 9

-(-a)=a

Answer 9

Cancellation Property of Opposites

Question 10

a+0=a 0+a=a

Answer 10

Identity Axiom for Addition

Question 11

a+(-a)=0 -a+a=0

Answer 11

Axiom of Additive Inverses

Question 12

-(a+b)=-a+(-b)

Answer 12

Property of the Opposite of a Sum

Question 13

a-b=a+(-b)

Answer 13

Definition of Subtraction

Question 14

a•1=a 1•a=a

Answer 14

Identity Axiom for Multiplication

Question 15

a•0=0 0•a=0

Answer 15

Multiplicative Property of Zero

Question 16

a(-1)=-a (-1)a=-a

Answer 16

Multiplicative Property of -1

Question 17

-a(b)=-ab a(-b)=-ab (-a)(-b)=ab

Answer 17

Property in Opposite Products

Question 18

a•1/a=1 1/a•a=1

Answer 18

Axiom of Multiplicative Inverses

Question 19

1/ab=1/a•1/b If a≠0, b≠0

Answer 19

Property of the Reciprocal of a Product

Question 20

a÷b=a•1/b If b≠0

Answer 20

Definition of Divsion

Question 21

Some smaller things

Answer 21

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