## 7. Properties of Addition and Multiplication

** **

**Theorem 7.1**:
(The Cancellation Property of Addition of Natural Numbers) If a, b,
and c are natural numbers and if

a + c = b + c
then

a = b.
**Definition 7.1**: If m and n are natural
numbers with m < n, then n - m is the m th predecessor of n.

**Theorem 7.2**: Let
S and T be two sets. Then

and

**Theorem 7.3**: Let
a, b, and c be natural numbers with b < a. Then

a - b = c
if and only if

c + b = a.
**Theorem 7.4**: For
any natural number a,

a - a = 0
**Theorem 7.5**:
**(Distributivity of Multiplication of Natural Numbers Across
Subtraction)** Let a, b, and c be natural numbers with c < b.
Then

a (b - c ) = ab - ac
**Theorem 7.6: (The
Integrity of Multiplication of Natural Numbers)** Let a and b be
natural numbers. ab = 0 if and only if one of a or b are 0.

**Theorem 7.7**: If
a and b are natural numbers then a + b __>__ a with equality if
and only if b = 0.

**Theorem 7.8**: If
a and b are non zero natural numbers, then

ab __>__ a
with equality if and only if b = 1.

**Theorem 7.9**: If
a, b, and c are nonzero natural numbers with b __<__ c, then

ab __<__ ac.
**Theorem 7.10: (The
Cancellation Property for Multiplication of Natural Numbers)** If
a, b, and c are natural numbers with a not 0 and ab = ac, then b = c.

8. The Division Algorithm
and the Fundamental Theorem of Arithmetic