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# Axioms (expounded)

TermDefinition
Closure prop. of Addition For any a and b in R, the sum a+b is in R.
Closure prop. of Multiplication For any a and b in R, the product a・b is in R.
Associative prop. of Addition For any a, b, and c in R, (a + b) + c = a + (b + c).
Associative prop. Multiplication For any a, b, and c in R, (a ・ b) ・ c = a ・ (b ・ c).
Commutative prop. of Addition For any a and b in R, a + b = b + a.
Commutative prop. of Multiplication For any a and b in R, a ・ b = b ・ a.
Distributive prop. of Multiplication over Addition For any a, b, and c in R, (a + b) ・ c = (a ・ c) + (b ・ c).
Multiplicative Identity There is a unique number 1 in R such that for any a in R, a ・ 1 = a.
Reflexive prop. of Equality For any a in R, a = a.
Symmetric prop. of Equality For any a and b in R, if a = b, then b = a.
Transitive prop. of Equality For any a, b, and c in R, if a = b and b = c, then a = c.
Addition prop. of Equality For any a, b, and c in R. if a = b, then a + c = b + c.
Multiplication prop. of Equality For any a, b, and c in R. if a = b, then a ・ c = b ・ c.
Created by: natureworld