24 terms

Axiom 1.1

(i) m + n = n + m

(ii) (m + n) + p = m + (n + p)

(iii) m(n+ p) = mn + mp

(iv) m**n = n**m

(v) (m**n) ** p = m ** (n**p)

(ii) (m + n) + p = m + (n + p)

(iii) m(n+ p) = mn + mp

(iv) m

(v) (m

Commutativity of addition

m + n = n + m

Associativity of addition

(m + n) + p = m + (n + p)

Distributitivity

m(n+ p) = mn + mp

Commutativity of multiplication

m**n = n**m

Associativity of multiplication

(m**n) ** p = m ** (n**p)

Axiom 1.2

(identity element for addition)

(identity element for addition)

There exists an integer 0 such that whenever m ∈ Z, m+0 = m.

Axiom 1.3

(identity element for multiplication)

(identity element for multiplication)

There exists an integer 1 such that 1 ≠ 0 and whenever m ∈ Z, m*1 = m.

Axiom 1.4

(additive inverse)

(additive inverse)

For each m ∈ Z, there exists an integer, denoted by m, such that m + (-m) = 0.

Axiom 1.5

(cancellation)

(cancellation)

Let m, n, and p be integers. If m**n = m**p and m ≠ 0, then n = p.

Reflexivity

m = m

Symmetry

If m = n then n = m

Transistivity

If m = n and n = p then m = p

Replacement

If m = n, then n can be substituted for m in any statement without changing its meaning

Divisibility

n|m means m is divisible by n.

Proposition 1.10

Let m, x₁, x₂ ∈ Z. If m, x₁, x₂ satisfy the equations m+ x₁= 0 and m+x₂ = 0, then x₁ = x₂ .

Proposition 1.12

Let x ∈ Z. If x has the property that for each integer m, m+x = m, then x = 0.

Proposition 1.13

Let x ∈ Z. If x has the property that there exists an integer m such that m+x = m, then x = 0.

Proposition 1.18

Let x ∈ Z. If x has the property that for all m ∈ Z, mx = m, then x = 1.

Proposition 1.19

Let x ∈ Z. If x has the property that for some nonzero m ∈ Z,

mx = m, then x = 1.

mx = m, then x = 1.

Proposition 1.23

Given m, n ∈ Z there exists one and only one x ∈ Z such that m + x = n

Proposition 1.26

Let m, n ∈ Z. If mn = 0, then m = 0 or n = 0.

Subtraction

m- n is defined to be m + (-n)

Proposition 1.27

(i) (m - n) + (p - q) = (m+ p) - (n+q).

(ii) (m - n) - (p - q) = (m+q) - (n+ p).

(iii) (m - n)(pq) = (mp+nq) - (mq+np).

(iv) m - n = p - q if and only if m + q = n + p.

(v) (m - n)p = mp - np.

(ii) (m - n) - (p - q) = (m+q) - (n+ p).

(iii) (m - n)(pq) = (mp+nq) - (mq+np).

(iv) m - n = p - q if and only if m + q = n + p.

(v) (m - n)p = mp - np.