Chapter 1: Integers

STUDY
PLAY
Axiom 1.1
(i) m + n = n + m
(ii) (m + n) + p = m + (n + p)
(iii) m(n+ p) = mn + mp
(iv) mn = nm
(v) (mn) p = m (np)
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
mn = nm
Associativity of multiplication
(mn) p = m (np)
Axiom 1.2
(identity element for addition)
There exists an integer 0 such that whenever m ∈ Z, m+0 = m.
Axiom 1.3
(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)
For each m ∈ Z, there exists an integer, denoted by m, such that m + (-m) = 0.
Axiom 1.5
(cancellation)
Let m, n, and p be integers. If mn = mp 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.
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.