## 6. Arithmetic

Definition 6.1: Let n, m be natural numbers. n + m is the m th successor of n.

Definition 6.2: Let S and T be two sets. If

then we say that they are disjoint.

Theorem 6.1: Let S and T two disjoint sets. Then

Theorem 6.2: It is possible to produce any number of pairwise disjoint sets of any size.

Theorem 6.3: (The Commutative Property for Addition of Natural Numbers) Let a and b be natural numbers. Then

a + b = b + a

Theorem 6.4: (The Associative Property for Addition of Natural Numbers) Let a, b, and c be natural numbers. Then

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

Theorem 6.5: (The Identity Property for Addition of Natural Numbers) Let n be a natural number. Then

n + 0 = n

Definition 6.3: Let n, m be natural numbers. nm is obtained by adding n to itself m times.

Convention: Order of operations: If you are combining numbers with addition and multiplication, we will agree that, unless there are parentheses, we will perform multiplications before we perform additions. If there are parentheses, they tell us to do what's inside parentheses first.

Theorem 6.6: Let M and N be two sets with #(M ) = m and #(N ) = n. Then

#(M x N ) = mn.

Theorem 6.7: (The Commutative Property for Multiplication of Natural Numbers): Let a and b be two natural numbers. Then

ab = ba

Theorem 6.8: (The Associative Property for Multiplication of Natural Numbers): Let a, b, and c be natural numbers. Then

a (bc ) = (ab)c

Theorem 6.9: (The Identity Property for Multiplication of Natural Numbers): Let n be a natural number. Then

(1)(n) = n

Theorem 6.10: ( The Distributive Property for Natural Numbers): Let a, b, and c be natural numbers. Then

a (b + c ) = ab + ac

7. Properties of Addition and Multiplication