Lecture16Sets - CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 16 — CS 2603 Applied Logic for

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Unformatted text preview: CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 16 — CS 2603 Applied Logic for Hardware and Software A Little Bit of Set Theory CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 What Is a Set? ¡ Axiomatic approach ¢ Postulate the existence of 9 Empty set — a set with no elements 9 Infinite set — a set with an infinite number of elements ¢ Define some set operations ¢ Build all other sets from this starting point ¡ Naïve approach ¢ A set is a collection of elements ¢ There is some way to determine whether or not an element resides in a set ¢ Avoid contradictions by being specific about the universe of discourse path we’ll follow CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 Notation for Sets — Explicit Enumeration ¡ {2, 3, 5, 7, 11} ¢ 2 ∈ {2, 3, 5, 7, 11} — stylized epsilon denotes “element of” ¢ 5 ∈ {2, 3, 5, 7, 11} ¢ 1 ∉ {2, 3, 5, 7, 11} — x ∉ A ↔ ¬ (x ∈ A) ¢ “Long Tall Sally” ∉ {2, 3, 5, 7, 11} ¡ {{“Wha’d I Say”, “Nadine”}, {“Peer Gynt”, “Moonlight Sonata”, “Finlandia”}} = A ¢ {“Wha’d I Say”, “Nadine”} ∈ A ¢ “Moonlight Sonata” ∉ A ¡ { } — the empty set, which has no elements none … nada … the number of elements in { } is zero ¢ 3 ∉ { } ¢ x ∉ { } — no matter what x stands for ¢ ∅ ≡ { } — stylized Greek letter phi denotes empty set ¢ ( ∅ ∈ { }) = True, or ( ∅ ∈ { }) = False ? False There is NOTHING in { } CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Notation for Sets — Set Comprehension ¡ { x | x ∈ {2, 3, 5, 7, 11} ∧ x ≥ 4} ¢ {5, 7, 11} ¡ { x + x | x ∈ {2, 3, 5, 7, 11} ∧ x ≥ 3 ∧ x < 11} ¢ {6, 10, 14} ¡ {f x | x ∈ A, p x} ¢ Denotes set whose elements have the form (f x), where x comes from A and (p x) has the value True ¡ To avoid contradictions ¢ Predicate (or context) specifies universe of discourse ¢ Examples of invalid set comprehensions 9 {X | X is a set} 9 {X | X ∉ X} 9 Universe of discourse is missing in these examples CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma...
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This note was uploaded on 04/08/2008 for the course CS 2603 taught by Professor Rexpage during the Spring '08 term at The University of Oklahoma.

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Lecture16Sets - CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 16 — CS 2603 Applied Logic for

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