Countable - Chapter 1 Chapter Set Theory Limits of...

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Chapter 1 Chapter 1 Set Theory Set Theory Limits of Computation Clayton Johnson Edna Reiter
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Limits of Computation by Johnson & Reiter 2 Sets A set is a collection of members or elements A set is said to contain its members There must always be an underlying universal set U
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Limits of Computation by Johnson & Reiter 3 Set Notation The elements of a set are listed in braces: • order doesn’t matter • repetition doesn’t matter {1, 2, 3, 4} {a, b, c} { , , }
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Limits of Computation by Johnson & Reiter 4 Set Notation with Ellipses Ellipses can be used in set notation once a pattern of membership has been established {1, 2, 3, …} {…, -3, -2, -1} {…, -2, -1, 0, 1, 2, …}
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Limits of Computation by Johnson & Reiter 5 Peano’s Notation The elements of a set can also be specified by using Peano’s notation: S = {x : x has property P} {x : x is a prime number} {x : x is a perfect square}
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Limits of Computation by Johnson & Reiter 6 Functions A B w x y z a b c d f :A B A function or mapping from set A to set B is a subset of A × B such that each x A is associated with a unique y B.
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Limits of Computation by Johnson & Reiter 7 Terminology of Functions For f :A B and f(x) = y, where x A and y B: A is the domain of f B is the codomain of f y is the image of x under f x is the preimage of y under f {y | y is an image under f } is the range of f
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Limits of Computation by Johnson & Reiter 8 Injective functions A B w x y z a b c f :A B A function f is injective (or one-to-one ) if preimages are unique. domain and range are the same size
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Limits of Computation by Johnson & Reiter 9 Surjective functions A B x y z a b c d f :A B A function f is surjective (or onto ) if each y B has a preimage codomain and range are the same size
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Limits of Computation by Johnson & Reiter 10 Bijective functions A B w x y z a b c d f :A B A function f is bijective (or a correspondence ) if it is both surjective and injective domain and codomain are the same size
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Countable - Chapter 1 Chapter Set Theory Limits of...

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