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Level 161

Axioms of Real Numbers


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a+b=b+a
Commutative Property of Addition
a*b=b*a
Commutative Axiom of Multiplication
(a+b)+c=a+(b+c)
Associative Property of Addition
(a*b)*c=a*(b*c)
Associative Axiom of Multiplication
a(b+c)=ab+ac
Distributive Property
a(b-c)=ab-ac
Distributive Axiom of Multiplication over subtraction
a-b=a+-b
Definition of Subtraction
a/b= a*1/b
Definition of Division
x+0=x
Additive Identity Axiom
x*1=x
Multiplicative Identity Axiom
x+(-x)=0
Additive Inverse Axiom
0*x=0
Multiplication property of 0
-1*x=-x
Multiplication Property of -1
Zero product
If ab=0 then either a=0 or b=0 or both
Transitive Axiom of equality
if x=y and y=z then x=z
Symmetric Axiom of Equality
if x=y then y=x
x=x
Reflexive Axiom of Equality
Addition Property of Equality
For all real numbers x, y, and z, if x = y, then x + z = y + z.
Multiplication Property of Equality
For all real numbers x, y, and z, if x = y, then xz = yz.
Closure Axiom under multiplication
if x=R and y=R, then x*y=R
Closure Axiom under Addition
if x=R and y=R, then x+y=R