Countability/Cardinality Help!

I don't understand how to do the following things,

(a) Let be a **surjection**. Show that if X is an **infinite countable** set then Y is either **countable or finite**. (set A is countable defined as |A| = |**N**|)

(b) By enumerating elements, or otherwise, show that the set of all ordered pairs of positive integers is countable.

(c) Deduce that the set of positive rationals is countable.

(d) Show that the set of all subsets of is uncountable [argue by contradiction]

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I don't understand how to do this stuff, any help is much appreciated.