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 r_k=r_k+1 * q_k+2 , r_k = 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