Discrete Math , Second series , 7 th Problem Set ( August 1 ) REU 2003 Instructor : László Babai Scribe
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چکیده
Definition 0.1. S is a countably infinite set if there exists a 1-1 correspondence between S and N. Exercise 0.2. Prove that the following sets are countably infinite: Q, Zn, Qn, the set of algebraic numbers. Definition 0.3. A set S has the cardinality of continuum if there exists a 1-1 correspondence between S and [0, 1]. Definition 0.4. Let S be a set. The power set of S, denoted 2S , is the set of all subsets of S. Exercise 0.5. Prove that the following sets have the cardinality of continuum: R, Rn, C[0, 1] (set of all continuous functions defined on [0, 1]), Sym(Z) (set of all permutations of Z), 2Z. Definition 0.6. We say that the cardinality of a set T is at least as big as the cardinality of a set S, i.e. card(S) ≤ card(T ), if there exists an injective function from S to T . Exercise 0.7. Prove: If card(S) ≤ card(T ) and card(T ) ≤ card(S) then card(S) = card(T ), i.e. there exists a bijection between S and T . Theorem 0.8 (Cantor). card(2S) > card(S), i. e., there is no 1-1 correspondence between S and its power set 2S.
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تاریخ انتشار 2003