UW-Madison CS 240 - Lecture 5 - Proofs (8 pages)

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Lecture 5 - Proofs



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Lecture 5 - Proofs

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Lecture Notes


Pages:
8
School:
University of Wisconsin, Madison
Course:
Cs 240 - Introduction to Discrete Mathematics

Unformatted text preview:

CS Math 240 Introduction to Discrete Mathematics 2 3 2011 Lecture 5 Proofs Instructor Dieter van Melkebeek Scribe Dalibor Zeleny DRAFT Up until now we have been introducing mathematical notation to capture concepts such as propositions implications predicates and sets We need this machinery in order to be able to argue properties of discrete structures in a rigorous manner As we were introducing new concepts we stated various facts and gave proofs of some of them but we were not explicit about what a good proof should look like Today we start discussing what constitutes a valid proof of a proposition and give some guidelines for writing proofs 5 1 Proofs We briefly mentioned what proofs were in the first lecture Let s repeat some of this discussion and make our definition of a proof more precise Definition 5 1 A proof of a proposition P is a chain of logical deductions ending in P and starting from some set of axioms Our definition of a proof mentions axioms and logical deductions both of which require further consideration Let s discuss them one by one 5 1 1 Axioms Axioms are statements we take for granted and do not prove The set of axioms we use depends on the area we work in For geometry we would use Euclid s five axioms of geometry Another set of axioms are the ZFC axioms the abbreviation stands for Zermelo Fraenkel and the axiom of Choice which form the basis of all set theory However both of these sets of axioms are small and proving any substantial result starting just from those axioms requires a significant amount of work Thus such sets of axioms are more suitable for a course on logic than a course on discrete structures In this course we use a much larger set of axioms because our focus is on proof techniques and their applications to discrete structures Thus we will consider any familiar fact from math at the level of high school as an axiom If you are unsure whether you can take something for granted on an assignment just ask 5 1 2 Logical Deductions



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