Axioms for Real Numbers

Question 1

commutative axiom

Answer 1

a+b=b+a
ab+ba

Question 2

associative axiom

Answer 2

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

Question 3

distributive axiom

Answer 3

a(b+c)=ab+ac

Question 4

reflexive axiom

Answer 4

a=a

Question 5

symmetric axiom

Answer 5

if a=b, then b=a

Question 6

Transitive Axiom

Answer 6

If a = b and b = c then a = c .
If a > b and b > c then a > c

Question 7

Addition Axiom

Answer 7

If a = b then a + c = b + c
If a > b then a + c > b + c

Question 8

Multiplication Axiom

Answer 8

If a = b then ac = bc
If a > b and c > 0 then ac > bc

Question 9

Existence of Additive Inverse

Answer 9

For every real number a, there exists a real number -a such that
a + (- a) = (- a) + a = 0

Question 10

Existence of Multiplicative Inverse

Answer 10

For every non-zero real number a, there exists a real number 1/a such that a(1/a) = 1 .

Question 11

Existence of Additive Identity

Answer 11

For every real number a, a + 0 = a

Question 12

Existence of Multiplicative Identity

Answer 12

for every real number a, a * 1 = 1 * a=a

Question 13

Trichotomy Axiom

Answer 13

For every real numbers a and b, exactly one of the following is true: a = b, a > b or a < b