Key Properties and Laws in Real Number Algebra

ax 1 = a

This represents the Identity Property of Multiplication.

If axb=axc and a ≠ 0, then b = c

This is known as the Cancellation Law of Multiplication.

a = a

This is an example of the Reflexive Property.

a + b = b + a

This illustrates the Commutative Property of Addition.

a + (-a) = 0

This represents the Inverse Property of Addition.

ax = 1 (if a ≠ 0)

This is the Definition of Division.

a - b = a + (-b)

This is the Definition of Subtraction.

For any real numbers a and b, exactly one of these is true: a < b, a = b, or a > b.

This is known as the Trichotomy Property.

If a < b, then a + c < b + c

This is the Addition Property of Order.

(a + b) + c = a + (b + c)

This illustrates the Associative Property of Addition.

ax(b + c) = (axb) + (axc)

This is the Left Distributive Property.

If a < b and c > 0, then a × c < b × c

This is the Multiplication Property of Order.

(axb) × c = ax(b × c)

This is the Associative Property of Multiplication.

If ab, then b = a.

This is the Symmetric Property.

If ab, then b < a.

This is a statement reflecting the properties of inequalities.

Since a ∈ R and b ∈ R, a × b ∈ R.

This is the Closure Property of Multiplication.

axb = bxa

This is the Commutative Property of Multiplication.

If a < b and b < c, then a < c

This is the Transitive Property of Order.

a + 0 = a

This is the Identity Property of Addition.

Since a ∈ R and b ∈ R, a + b ∈ R.

This is the Closure Property of Addition.

a + b = ax (if b = 0)

This is a statement regarding the properties of addition.

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

This is the Transitive Property of Equality.