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What’s wrong with this proof?

What’s wrong with this proof?. If you figure it out, don’t call it out loud – let others ponder it as well. Let a and b be non-zero such that a = b Multiply both sides by a a 2 = ab Subtract b 2 a 2 - b 2 = ab-b 2 Factor both sides ( a - b )( a + b ) = b ( a - b )

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What’s wrong with this proof?

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  1. What’s wrong with this proof? If you figure it out, don’t call it out loud – let others ponder it as well. • Let a and b be non-zero such that a = b • Multiply both sides by aa2 = ab • Subtract b2a2-b2 = ab-b2 • Factor both sides (a-b)(a+b) = b(a-b) • Divide by (a-b) a+b = b • Since a = b, we replace b with ab+b = b • Combine terms 2b = b • As b is non-zero, divide it out 2 = 1 Q.E.D. (Latin for “which was to be proven”)

  2. CS 202: Discrete Math Spring 2007 Aaron Bloomfield

  3. So…. What is it? • Discrete mathematics … is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity (wikipedia) • In other words, dealing with integer things (sets, logic, proofs, etc.) and not continuous things (calculus, functions, etc.)

  4. Why discrete math? • It forms the basis for computer science: • Sequences • Digital logic (how computers compute) • Algorithms • Assuring computer correctness • Probability and gambling (really!) • Etc. • Like how “regular” math forms the basis for science

  5. Sequences in Nature 13 8 5 3 2 1

  6. Proofs • How do you know something is correct? • How do you know when something is not correct? • Such as showing that 2=1? • How do you think logically? • How do you think to solve problems?

  7. What’s wrong with this proof? If you figure it out, don’t call it out loud – let others ponder it as well. • Let a and b be non-zero such that a = b • Multiply both sides by aa2 = ab • Subtract b2a2-b2 = ab-b2 • Factor both sides (a-b)(a+b) = b(a-b) • Divide by (a-b) a+b = b • Since a = b, we replace b with ab+b = b • Combine terms 2b = b • As b is non-zero, divide it out 2 = 1 Q.E.D. (Latin for “which was to be demonstrated”)

  8. Why am I here?

  9. Course objectives • Logic: Introduce a formal system (propositional and predicate logic) which mathematical reasoning is based on • Proofs: Develop an understanding of how to read and construct valid mathematical arguments (proofs) and understand mathematical statements (theorems), including inductive proofs. Also, introduce and work with various problem solving strategies and techniques • Counting: Introduce the basics of integer theory, combinatorics, and counting principles, including a brief introduction to discrete probability • Structures: Introduce and work with important discrete data structures such as sets, relations, sequences, and discrete functions • Applications: Gain an understanding of some application areas of the material covered in the course

  10. Aaron Bloomfield Office: Olsson 228D Office hours: M/W 4:00-5:30 Email: Who am I?

  11. Who are they? • We will have a number of teaching assistants • Who will be holding office hours • This will be posted on the website in a few days • Office hours start next week

  12. Going over the syllabus…

  13. Textbook (Required) • Susanna Epp • Discrete Mathematics with Applications, 3rd edition • ISBN 0534359450 • Sorry about the price!

  14. Other Issues • Privacy • Keeping the class interesting • Most of you know that I like humor • But most of you have seen my humor… • So we’ll be seeing puzzles and brain teasers this semester • And you get to contribute!

  15. Where did the money go? Three people check into a hotel. They pay $30 to the manager and go to their room. The manager suddenly remembers that the room rate is $25 and gives $5 to the bellboy to return to the people. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has $2, totaling $29. Where is the missing $1?

  16. Feedback • It’s a very good thing! • Feel free to leave us feedback • Can be done anonymously, if you wish • Via the Toolkit or the CS dept website • It’s hard for the instructors to know what the students think of the course…

  17. The Study • How does one make a discrete math class more interesting? • By adding: • “Nifty” problems • Application examples • Physical demonstrations • Interactive demonstrations • Brain teasers • This is one reason why there are labs recitation sections.

  18. Scheduling Issues • This room has 94 seats • One of the lab sections (11 a.m.) has 42 seats • I will course action people in starting next Monday • I am planning on taking a survey of who wants to get into which section • For now, come to the 1:00 recitation if you can

  19. The survey • This will help me with the recitation scheduling, background material coverage, etc.

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