Algebra 2 The Field Axioms

Commutativity

Addition and multiplication of real numbers are ___ operations. That is, if x and y are real numbers, then x + y and y + x are equal to each other, xy and yx are equal to each other.

Associativity

Addition and multiplication of real numbers are ___ operations. That is, if x, y, and z are real numbers, then (x + y) + z and x + (y + z) are equal to each other. (xy)z and x(yz) are equal to each other.

Transitivity

For equality: If x = y and y = z, then x = z. For order: If x < y and y < z, then x < z. If x > y and y > z, then x > z.

Symmetry

If x = y, then y= x.

Trichotomy

If x and y are two real numbers, then exactly one of the following must be true: y < x y > x y = x

Reflexive

If x is a real number, then x=x.

Distributivity

Multiplication ___over addition. That is, if x, y, and z are real numbers, then x(y + z) and xy + xz are equal to each other.

Inverses

{real numbers} contains: a unique additive ___ for every real number x. (Meaning that every real number x has a real number -x such that x + (-x) = 0.) A unique multiplicative ___ inverse for every real number x except zero. (Meaning that every non-zero number x has a real number 1/x such that x•1/x = 1.)

Identity elements

{real numbers} contains: a unique identity element for addition, namely 0. (Because x + 0 = x for any real number x.) a unique identity element for multiplication, namely 1. (Because x•1 = x for any real number x.)

Closure

{real numbers} is closed under addition and under multiplication. That is, if x and y are real numbers, then x + y is a unique real number, xy is a unique, real number.