Lecture1 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 1 Dr. Ted Ralphs IE417 Lecture 1 1 Reading for This Lecture Suggested references Solow , How to Read and Do Proofs Bittinger , Logic and Proof Velleman , How to Prove It IE417 Lecture 1 2 Mathematical Proof Techniques IE417 Lecture 1 3 Mathematical Systems Elements of a mathematical system A universal set A set of relations A set of operations A set of axioms To these given elements, we can add Definitions Theorems IE417 Lecture 1 4 Example: The Natural Numbers Counting Axioms (Peano) : 1 is a natural number. For each natural number x , there exists exactly one natural number, called the successor of x , and denoted x For all natural numbers x , x 6 = 1 If x and y are natural numbers such that x = y , then x = y Axiom of Induction : If S is a set of natural numbers, then if 1 S , and x S , then x S then S contains all of the natural numbers. Question : Does belong to the natural numbers? Bonus Question : What is a naturaaal property of the integers that cannot be proved using only the Peano axioms? IE417 Lecture 1 5 Definitions, Theorems, Etc. A definition is simply an abbreviation or shortcut for a longer phrase. Example : A set is said to be convex if . . . From then on, we can simply use the term convex set instead of spelling out the property itself....
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This note was uploaded on 08/06/2008 for the course IE 417 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

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Lecture1 - Advanced Mathematical Programming IE417 Lecture...

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