set theory l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Set Theory PowerPoint Presentation
Download Presentation
Set Theory

Loading in 2 Seconds...

play fullscreen
1 / 11

Set Theory - PowerPoint PPT Presentation


  • 187 Views
  • Uploaded on

Set Theory. Relations, Functions, and Countability. Relations. Let B ( n ) denote the number of equivalence relations on n elements. Show that B(n) ≤ . Show that B(n) ≤ n!. Show that B(n) ≥ 2 n−1 . . Bell numbers. Functions and Equivalence Relations. Remark

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Set Theory' - sokanon


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
set theory

Set Theory

Relations, Functions, and Countability

relations
Relations
  • Let B(n) denote the number of equivalence relations on n elements.
  • Show that B(n) ≤ .
  • Show that B(n) ≤ n!.
  • Show that B(n) ≥ 2n−1 .

Bell numbers

functions and equivalence relations
Functions and Equivalence Relations

Remark

Equivalence relation is a relation that is reflexive, symmetric, and transitive

  • Suppose that:
  • Is a function?
  • Which of the following is an equivalence relation?

where Δ(x, y) denotes the Hamming distance of x and y,

cardinality
Cardinality
  • A and Bhave the same cardinality(written |A|=|B|) iff there exists a bijection (bijective function) from A to B.
  • if |S|=|N|, we say S is countable. Else, S is uncountable.
cantor s theorem
Cantor’s Theorem
  • The power set of any set A has a strictly greater cardinality than that of A.
  • There is no bijection from a set to its power set.

Proof

  • By contradiction
countability
Countability
  • An infinite set A is countably infinite if there is a bijection f: ℕ →A,
  • A set is countable if it finite or countably infinite.
countable sets
Countable Sets
  • Any subset of a countable set
  • The set of integers, algebraic/rational numbers
  • The union of two/finnite sum of countable sets
  • Cartesian product of a finite number of countable sets
  • The set of all finite subsets of N;
  • Set of binary strings
uncountable sets
Uncountable Sets
  • R, R2, P(N)
  • The intervals [0,1), [0, 1], (0, 1)
  • The set of all real numbers;
  • The set of all functions from N to {0, 1};
  • The set of functions N → N;
  • Any set having an uncountable subset
transfinite cardinal numbers
Transfinite Cardinal Numbers
  • Cardinality of a finite set is simply the number of elements in the set.
  • Cardinalities of infinite sets are not natural numbers, but are special objects called transfinite cardinal numbers
  • 0:|N|, is the first transfinite cardinal number.
  • continuum hypothesis claims that |R|=1, the second transfinite cardinal.
one to one correspondence
One-to-One Correspondence
  • Prove that (a, ∞) and (−∞, a) each have the same cardinality as (0, ∞).
  • Prove that these sets have the same cardinality: (0, 1), (0, 1], [0, 1], (0, 1) U Z, R
  • Prove that given an infinite set A and a finite set B, then |A U B| = |A|.