##### Properties of Addition

A1. *Associativity*: If a,b,c**Z**, then (a+b)+c = a+(b+c).

A2. *Commutativity*: If a,b**Z**, then a+b = b+a.

A3. *Identity*: If a**Z**, then a+0 = 0+a = a.

A4. *Inverses*: If a**Z**, then there exists a unique integer b such that a+b = b+a = 0.
##### Properties of Multiplication

M1. *Associativity*: If a,b,c**Z**, then (a·b)·c = a·(b·c).

M2. *Commutativity*: If a,b**Z**, then a·b = b·a.

M3. *Identity*: If a**Z**, then a·1 = 1·a = a.

##### Distributive Law

Addition and multiplication in **Z** are related by the *distributive
law* which asserts that if a,b,c**Z**, then a·(b+c) = a·b+a·c.

P0. Let a**Z**. Then exactly one of the following is true: a**P**, a = 0, -a**P**.

P1. If x,y**P**, then x=y**P**.

P2. If x,y**P**, then x·y**P**.

##### V. Principle of Mathematical Induction

Let S be a nonempty subset of the set of positive integers **P**. Assume that
(a) 1S and
(b) whenever nS we have n+1S. Then S contains all positive integers.

##### Order

O1. Exactly on of the following holds: x<y, x = y, x>y.

O2. If x<y, y<z, then x<z.

O3. If x<y and z<w, then x+z < y+w.

O4a. If x < y and z > 0 then xz < yz.

O4b. If x < y and z < 0 then xz > yz.

**Integer Proposition 1**: If b is a positive integer. Then b is greater
than or equal to 1.

**Integer Corollary 2**: Let x and y be integers. If x > y, then
x __>__ y+1.

**Integer Corollary 3**: Let and b positive integers. Then there exists a positive integer n such that a __<__ nb.

**Integer Theorem 4**: Let a and b be nonzero integers. Then
a · b 0.

**Integer Proposition 5**: Let S be a finite, nonempty subset of **Z**. Then S has a largest and smallest element.

**Integer Theorem 6**: Let S be a nonempty (finite or infinite) subset of **N**. Then S contains a smallest element.

**Integer Theorem 7**(*Division Algorithm*): Let a be a nonnegative integer and let b be a positive integer. Then there exist natural numbers q and t such that 0 __<__ r < b and
a = bq + r.