What’s wrong with this proof?
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What’s wrong with this What’s wrong with this proof? proof? If you figure it out, don’t call it out If you figure it out, don’t call it out loud – loud – let others ponder it as well. let others ponder it as well. 1. 1. Let Let a a and and b b be non-zero such that be non-zero such that a = b a = b 2. 2. Multiply both sides by Multiply both sides by aa 2 = = ab ab 3. 3. Subtract Subtract b 2 a a 2 - b b 2 = = ab-b ab-b 2 4. 4. Factor both sides Factor both sides ( ( a a - - b b )( )( a a + + b b ) = ) = b b ( ( a a - - b b ) ) 5. 5. Divide by ( Divide by ( a-b a-b ) ) a+b a+b = = b b 6. 6. Since Since a = b a = b , we replace , we replace b with with a b+b = b b+b = b 7. 7. Combine terms Combine terms 2 b b = = b 8. 8. As As b is non-zero, divide it out is non-zero, divide it out 2 = 1 2 = 1 Q.E.D. (Latin for “which was to be proven”) Q.E.D. (Latin for “which was to be proven”)
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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 ) - PowerPoint PPT Presentation
Transcript of What’s wrong with this proof?
What’s wrong with this What’s wrong with this proof?proof?
If you figure it out, don’t call it out loud – If you figure it out, don’t call it out loud – let others ponder it as well.let others ponder it as well.
1.1. Let Let aa and and bb be non-zero such that be non-zero such that a = ba = b2.2. Multiply both sides by Multiply both sides by aa aa22 = = abab3.3. Subtract Subtract bb22 aa22--bb22 = = ab-bab-b22
4.4. Factor both sidesFactor both sides ((aa--bb)()(aa++bb) = ) = bb((aa--bb))5.5. Divide by (Divide by (a-ba-b)) a+ba+b = = bb6.6. Since Since a = ba = b, we replace , we replace bb with with aa b+b = bb+b = b7.7. Combine termsCombine terms 22bb = = bb8.8. As As bb is non-zero, divide it out is non-zero, divide it out 2 = 12 = 1
Q.E.D. (Latin for “which was to be proven”)Q.E.D. (Latin for “which was to be proven”)
CS 202: Discrete Math
Spring 2007
Aaron Bloomfield
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.)
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
Sequences in Nature
1385321
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?
What’s wrong with this What’s wrong with this proof?proof?
If you figure it out, don’t call it out loud – If you figure it out, don’t call it out loud – let others ponder it as well.let others ponder it as well.
1.1. Let Let aa and and bb be non-zero such that be non-zero such that a = ba = b2.2. Multiply both sides by Multiply both sides by aa aa22 = = abab3.3. Subtract Subtract bb22 aa22--bb22 = = ab-bab-b22
4.4. Factor both sidesFactor both sides ((aa--bb)()(aa++bb) = ) = bb((aa--bb))5.5. Divide by (Divide by (a-ba-b)) a+ba+b = = bb6.6. Since Since a = ba = b, we replace , we replace bb with with aa b+b = bb+b = b7.7. Combine termsCombine terms 22bb = = bb8.8. As As bb is non-zero, divide it out is non-zero, divide it out 2 = 12 = 1
Q.E.D. (Latin for “which was to be demonstrated”)Q.E.D. (Latin for “which was to be demonstrated”)
Why am I here?
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
Who am I?
• Aaron Bloomfield– Office: Olsson 228D– Office hours: M/W
4:00-5:30– Email:
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
Going over the syllabus…
Textbook (Required)
• Susanna Epp
• Discrete Mathematics with Applications, 3rd edition– ISBN 0534359450
• Sorry about the price!
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!
Where did the money go?Where did the money go?
Three people check into a hotel. They pay $30 Three people check into a hotel. They pay $30 to the manager and go to their room. The to the manager and go to their room. The manager suddenly remembers that the room manager suddenly remembers that the room rate is $25 and gives $5 to the bellboy to return rate is $25 and gives $5 to the bellboy to return to the people. On the way to the room the to the people. On the way to the room the bellboy reasons that $5 would be difficult to bellboy reasons that $5 would be difficult to share among three people so he pockets $2 share among three people so he pockets $2 and gives $1 to each person. Now each person and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has $2, totaling each, totaling $27. The bellboy has $2, totaling $29. Where is the missing $1?$29. Where is the missing $1?
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…
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.
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
The survey
• This will help me with the recitation scheduling, background material coverage, etc.
