Question: Note: All information is given, if you need any more than comment. 1) (a) Define f: Z - Z by setting f (x) = x2

Note: All information is given, if you need any more than comment.

(a) Define f: Z - Z by setting f (x) = x2 + 1 for all x E Z. (i) Let X, = {-1,0,1}. Is it true that If (X,)| > IX,|? [2 marks] (ii) Let X2 = {1,2,3}. Is it true that If (X2)| > |X2|? [2 marks] Explain your answers. (b) Let g: A - B such that |g (X)| > [X| for all finite subsets X C A. Prove that g is injective. [4 marks]Given mutually disjoint events 11,142, m, An, prove that P (0 Ai) = :1: P040 11 Hint: Use mathematical induction. Please use 001) instead of P(n) for the proposition in your induction so as not to be mixed up with P the probability symbol

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