## 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