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Set Theory

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

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Set Theory

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  1. Set Theory Relations, Functions, and Countability

  2. 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

  3. 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,

  4. 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.

  5. 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

  6. 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.

  7. 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

  8. Diagonal Argument

  9. 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

  10. 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.

  11. 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|.

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