Landing : Athabascau University

MATH 309 - Unit 1

  • Public
By Xing Li June 12, 2019 - 12:01pm

Unit 1 Integer

(net time: 190 min)

equality 8 axioms: A4M3D1 can lead to P3 of uniqueness

              A1. associativity: (k + m) + n = k + (m + n)

A2. commutativity: m+n=n+m

A3. additive identity: 0 + n=n

A4. additive inverse: n + (-n) = 0

M3. multiplicative identity: 1*n = n

D. distributivity: n(k + m) = nk + nm = kn + mn = (k + m)n:

P4. n = -(-n)

P11. 0*n = 0

P12. (-1) n=-n, (-m) n=m (-n) = - mn


order 3 axioms; Peano axiom

              O1: m>0, n>0 then mn>0, m+n>0

O2 = trichotomy

              cancellation law: m<n ó k+m < k+n

              transitivity: If k < m and m < n; then k < n

P19.       c. If m < n and k > 0; then mk < nk:

d. If m < n and k < 0; then mk > nk:

e. If b < k and m < n; then b + m < k + n:


the first law of disjunction: If we consider the statement P or Q, and if P is not true (i.e., does not

hold), then we conclude that Q is true (holds).

the well-ordering axiom: Every nonempty subset of Z+ contains a smallest element

Division Algorithm: a=qb+r unique expression

              P29. if d|a and d|b then d|r

Euclidean Algorithm: a=bq1+r1; b=r1*q2+r2 until rk=rk+1 * qk+2 , rk = gcd (a,b)

Theorem P49 ab=gcd(a,b) * lcm(a, b)

Principle of Mathematical Induction, deducted from Peano axiom.

second principle of induction, k<n true -> n true

Fundamental theorem of arithmetic: prime decomposition is possible and unique


These comments are moderated. Your comment will not be visible unless accepted by the content owner.

Only simple HTML formatting is allowed and any hyperlinks will be stripped away. If you need to include a URL then please simply type it so that users can copy and paste it if needed.