hw8 - (1) For any set S define P (S) to be the set of all...

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Unformatted text preview: (1) For any set S define P (S) to be the set of all subsets of S. for example, if S = {a, b} then P (S) = {, {a}, {b}, {a, b}}. Let A be a finite set. Show that |P (A)| = 2|A| . Hint: Let A = {x1 , . . . , xn }. Represent a subset S of A by a sequence of 0s and 1s of length n such that the i-th element in the sequence is 1 if xi S and is 0 if xi S. / (2) Let S be an infinite set. Prove that |S| |N|. (3) Let S = [0, 1) and T = [0, 1]. Let f : S T be given by f (x) = x and g : T S be given by g(x) = x/2. (a) Find SS , ST , S , TS , TT , T (b) give an explicit formula for a 1-1 and onto map h : S T coming from f and g using the proof of the Schroeder-Berenstein theorem. (4) Let S be infinite and s0 S. Prove that |S| = |S\{s0 }|. Hint: Construct a map from S to S\{s0 } similar to the map h from Problem (3). (5) Let S be the set of all subsets of N which contain exactly 5 elements. Prove that S is countable. (6) Let S = P (N) Show that |S| |R|. Hint: represent a subset A of N a sequence of 1s and 0s such that the n-th element of the sequence is 1 if n A and is 0 if n A. / 2 + y 2 = 1 in R2 . (7) Find the cardinality of the circle x 1 ...
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