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  3. 7 let x be a set and let px be...

Question: 7 let x be a set and let px be...

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(7) Let X be a set, and let P(X) be its power set. Let f:Xx - P(X) be an arbitrary map. Define a subset Sf of X as follows: (i) (4 pts) Show that Sf is not in the image of f (ii) (1 pt) Conclude that f is not surjective. (iii) (1 pt) Show that there is no surjective map X → p(X). (iv) (1 pt) Conclude that there is no bijective map X → P(X). Hint. Suppose there is an element r E X such that S- f(o). There are two possibilities: either ro belongs to S or zo does not belong to Sf. Show that actually none of these two possibilities occurs, and therefore obtain a contradiction.

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