2. Let X be an uncountable set and let Y be a countable set. (a) If f : X → Y is a function, prove that some element of Y has an uncountable pre-image, that is, show there exists y € Y with |N] < |f¯'(y)|- (b) Let Z be a subset of X. If Z is countable, prove that X \ Z is uncountable. (c) Let Z1, Z2, ..., Z, be subsets of X. If Z; is countable for i = 1,...,n, prove that (-) X \ Zi is uncountable.

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