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MacBook Air Is TIL leylu Top Hwa, Real Analysis, due 1/22/2020 O Recall Prove that that Qt denotes the set of positive rational nun Qt Qt is countably infinite, © Give an explicit example of sets such that for every nal An' Anti is infinite, A, A2, A3,... Antic An and Give an example of a surjective function fi IN-IN which is not a bijection. Also prove that any surjective function f. 61,2,.. n} {1, 2, 3,.. n} is a bijection. Prove that VT is not a rational number,

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28 29 3 Let us Consider the map film IN defined by fm) =sm if n ibodd in if n iberen 1. FCN) = N Hence it is surpective But I has two preimages 12 s. The mapp is not infective Consequently the map is not bijedine - For any insurrective map fi {1, 2,., n} {1, 2 org m} We must home n-number of images and since domain Set have only in element So their a images must be distimet in order to have n images. Hence the map is infective Consequently the app is birective -