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Test1 - 1 Russells Paradox If S is a set there are two...

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1 Russell’s Paradox If S is a set, there are two possibilities. S S or S S . Let G = { S S is a set & S S } . Let B = { S S is a set & S S } C1: If G G then G is a set & G G C2: If G G , then G G 2 Functions A function f A B ∋ ∀ x A f ( x ) ∈ 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 A B is injective (or one-to-one) If, x,y A, f ( x ) = f ( y ) x = y Definition 2 . A function f A 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, ǫ > 0 , δ > 0, such that y , if divides.alt0 x y divides.alt0 < δ then divides.alt0 f ( x ) − f ( y )divides.alt0 < ǫ . Definition 4 . A function f is not continuous at x when, ǫ > 0 , δ > 0 , y such that divides.alt0 x y divides.alt0 < δ , but that ǫ ≤ divides.alt0 f ( x ) − f ( y )divides.alt0 .
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