Set Theory - PowerPoint by 1aWl7i9n

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									          Set Theory

Relations, Functions, and Countability
Relations
• Let B(n) denote the number of equivalence relations on n elements.


• Show that B(n) ≤            .                           Bell numbers

• Show that B(n) ≤ n!.

• Show that B(n) ≥ 2n−1 .
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
• A and B have 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
• 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
• 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
•   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
Diagonal Argument
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
• 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

1.   Prove that (a, ∞) and (−∞, a) each have the same
     cardinality as (0, ∞).

2.   Prove that these sets have the same cardinality: (0, 1),
     (0, 1], [0, 1], (0, 1) U Z, R

3.   Prove that given an infinite set A and a finite set B,
     then |A U B| = |A|.

								
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