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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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Lecture1 - Advanced Mathematical Programming IE417 Lecture...

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