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Exploring Bell Numbers, Functions, and Equivalence Relations in Set Theory

This comprehensive guide delves into Bell numbers, functions, and equivalence relations in set theory. Understand the cardinality of relations and functions, prove equivalence relations, and explore countability concepts. Learn about Cantor's Theorem, countable sets, uncountable sets, transfinite cardinal numbers, and cardinality comparisons. Delve into proofs and the concepts of Diagonal Argument and One-to-One Correspondence.

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Exploring Bell Numbers, Functions, and Equivalence Relations in 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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