ITEC 420 2023fall ibarland

home—lectures—exams—hws—D2L—zoom (snow day)

Sets and cardinalities

home—lectures—exams—hws—D2L—zoom (snow day)

READING: Chpt.03

Previously, on itec420…

Sets, and set-operations — making new sets out of old ones. What was the concatenation of two languages defined as, again?
For two languages A,B, we say:

Recall lexicographic order.

Finishing up languages/sets

Kleene star: A*; Sigma*

define: "the set A closed under f":
  - ints closed under squaring (unary operator)
  - ints closed under * (binary operator)
  - ints *not* closed under square-rooting, nor division
  - general def'n, in ENGLISH (logic as exer. below)
       unary f : X→Y :  for any number n, f(n)
      (for a unary f, and separately for a binary f)

- Showing sets equal:
    compare:
    L₃  = strings over {a,b}* where every 'a' is followed by a 'b'
    vs
    L₃' = strings over {a,b}* which don't contain "aa"

   Show/argue :
      - L₃ ⊆  L₃': quick proof-by-contra
      - try other dir (careful!)

- review proof-by-induction: n(n+1)(2n+1)/6
    sum-of-squares formula

Countability

Def'n: a set S is countable if ∃f . f:ℕ→S, f is onto.¹
(If S is countably infinite, it's convenient to also require f to be 1:1 — a bijection, and hence f will be invertable.)

1. Def'n
2. Σ^* countable, (e.g. Σ = {0, 1, 2, …, 9}). Corollary: Java programs countable.
3. primes countable
4. ℤ countable
5. 0.01⋅ℕ = {0, 0.01, 0.02, …} countable
6. ℚ countable!!
7. ℝ not countable: sketch diagonalization argument

home—lectures—exams—hws—D2L—zoom (snow day)

This page licensed CC-BY 4.0 Ian Barland Page last generated

Please mail any suggestions (incl. typos, broken links) to ibarlandradford.edu