## 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