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Unformatted text preview: 1 Russellâ€™s Paradox If S is a set, there are two possibilities. S âˆˆ S or S âˆ‰ S . Let G = { S âˆ¶ S is a set & S âˆ‰ S } . Let B = { S âˆ¶ S is a set & S âˆˆ S } C1: If G âˆˆ G then G is a set & G âˆ‰ G C2: If G âˆ‰ G , then G âˆˆ G 2 Functions A function f âˆ¶ A â†’ B âˆ‹ âˆ€ x âˆˆ A â†’ f ( x ) âˆˆ B . f maps to f ( x ) . f ( x ) is the image of x under f . x must map to a single f ( x ) . Definition 1 . A function f âˆ¶ A â†’ B is injective (or onetoone) If, âˆ€ x,y âˆˆ A, f ( x ) = f ( y ) â†’ x = y Definition 2 . A function f âˆ¶ A â†’ B is surjective (onto) If, range ( f ) = B 3 Logic/Truth Tables False implies True. 4 Definitions Definition 3 . A function f is continuous at a point x if, âˆ€ Ç« > , âˆƒ Î´ > 0, such that âˆ€ y , if divides.alt0 x âˆ’ y divides.alt0 < Î´ then divides.alt0 f ( x ) âˆ’ f ( y )divides.alt0 < Ç« . Definition 4 . A function f is not continuous at x when, âˆƒ Ç« > , âˆ€ Î´ > , âˆƒ y such that divides.alt0 x âˆ’ y divides.alt0 < Î´ , but that Ç« â‰¤ divides.alt0â‰¤ divides....
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This note was uploaded on 01/23/2012 for the course MATH 3012 taught by Professor Costello during the Fall '08 term at Georgia Institute of Technology.
 Fall '08
 COSTELLO

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