1
Proof Must Have
• Statement of what is to be proven.
• "Proof:" to indicate where the proof starts
• Clear indication of flow
• Clear indication of reason for each step
• Careful notation, completeness and order
• Clear indication of the conclusion
Number Theory - Ch 3 Definitions
•Z--- integers
•Q- rational numbers (quotients of integers)
– r∈Q↔ ∃a,b∈Z, (r = a/b) ^ (b ≠0)
• Irrational = not rational
•R --- real numbers
• superscript of +--- positive portion only
• superscript of ---- negative portion only
• other superscripts: Zeven, Zodd , Q>5
• "closure" of these sets for an operation
Integer Definitions
• even integer
– n ∈Zeven ↔ ∃k ∈
∈∈
∈Z n = 2k
• odd integer
– n ∈Zodd ↔ ∃k ∈
∈∈
∈Z n = 2k+1
• prime integer (Z>1)
– n ∈Zprime ↔ ∀r,s∈Z+, (n=r*s) →(r=1)v(s=1)
• composite integer (Z>1)
– n ∈Zcomposite ↔ ∃r,s∈Z+, n=r*s ^(r≠1)^(s≠1)
