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# Axioms

### Axioms of addition and multiplication in R. Let a, b, and c denote any real #'s

TermDefinition
Closure Axiom for Addition a+b is a unique real number
Associative Axiom for Addition (a+b)+c=a+(b+c)
Commutative Axiom for Addition a+b=b+a
Identity Axiom for Addition There exists an element 0∈R such that for each a∈R, 0+a=a and a+0=a
Axiom of Additive Inverses There exists an element -a∈R for each a∈R, such that a+(-a)=0 and (-a)+a=0
Closure Axiom for Multiplication ab is a unique real number
Associative Axiom for Multiplication (ab)c=a(bc)
Commutative Axiom for Multiplication ab=ba
Identity Axiom for Multipication There exists an element 1∈R, 1≠0, such that for each a∈R, a·1=a and 1·a=a.
Axiom of Multiplicative Inverses There exists an element 1/a∈R for each nonzero a∈R such that 1/a·a=1 and a·1/a=1
Distributive Axiom of Multiplication with Respect to Addition a(b+c)=ab+bc and (b+c)a=ba+ca
Created by: 100006167281117