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Homework 03

# Homework 03 - X ⊆ Y with Y a ﬁnite set Use the previous...

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MATH 74 HOMEWORK 3 To see a world in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour. . . -William Blake Due Wednesday February 18th at 3:10pm. (1) Prove that | 2 X | = 2 | X | where 2 X is the power set of X . Hint: show that an element of 2 X is equivalent to a function X → { 0 , 1 } , so then the set of all such subsets is the same as the set of all functions X → { 0 , 1 } . (2) Prove that if A and B are countable, then A B is countable. (HINT: Think about the proof that the integers are countable, and think of the integers as two copies of the natural numbers.) (3) Prove that if A is uncountable and B is countable, then A - B is uncountable. Conclude that the irrational numbers (i.e. R - Q ) form an uncountable set. (4) Suppose that X is a non-empty finite set of cardinality n . Prove that there is a bijection X Y if and only if | Y | = n . (5) Let
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Unformatted text preview: X ⊆ Y with Y a ﬁnite set. Use the previous problem to show that (a) X = Y if and only if | X | = | Y | . (b) X ± Y if and only if | X | < | Y | . (6) Suppose that X and Y are non-empty ﬁnite sets of cardinality n . Prove that a map X → Y is a surjection if and only if it is an injection. (7) Let X and Y be ﬁnite sets (possibly empty). Prove that | X × Y | = | X | × | Y | . (8) Suppose that { 1 ,...,n } → X is a surjection. Prove that | X | ≤ n . (Hint: Do induction on n .) (9) Let f : X → Y be a function between metric spaces ( X,d ), ( Y,f ). Using the deﬁnition of continuity from class, prove that f is continuous if and only if the inverse image of every open set is open, i.e. for all open sets U ⊂ Y , f-1 U is an open set in X . 1...
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