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MATH 135 MIDTERM

MATH 135 MIDTERM. About Me :D. 2A Combinatorics & Optimization and Computer Science I love Math! (and Algebra too) Passionate about teaching Best quote that describes me: “People who don’t know me think I’m shy. People who know me wish I was.”

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MATH 135 MIDTERM

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  1. MATH 135 MIDTERM

  2. About Me :D • 2A Combinatorics & Optimization and Computer Science • I love Math! (and Algebra too) • Passionate about teaching • Best quote that describes me: “People who don’t know me think I’m shy. People who know me wish I was.” • Email – hockeyrahul@gmail.com (don’t ask)Facebook – brahul90

  3. 2010 Outreach Trip Summary Date Aug 20 – Sept 4 Location Cusco, Peru # Students 22 Project Cost $16,000 Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects ELT Classroom enables better job prospects More info @ studentsofferingsupport.ca/blog

  4. Agenda 1.1 The Language of Mathematics 1.2 Logic 1.3 Sets 1.4 Quantifiers 1.5 Proofs 1.6 Counterexamples 4.1 Mathematical Induction 4.2 Recursion 4.3 Binomial Theorem 2.1 Division Algorithm 2.2 Euclidean Algorithm What should I stress on?

  5. 1.1 The Language of Mathematics • Theorem: a major landmark • Proposition: a lesser result • Lemma: minor result • Corollary: follows immediately • Example: particular case • Algorithm: explicit procedure • Proof: mathematical argument

  6. 1.2 Logic • A statement or proposition is a sentence that is either TRUE or FALSE (Note: Questions are never statements). • Propositional Logic: • P AND Q is called the conjunction of P and Q. • P OR Q is called the disjunction of P and Q. • The negation of the statement P is denoted by NOT P. • Conditional statement: “If P, then Q” • Converse of “If P, then Q” is “If Q, then P”. • “P if and only if Q”

  7. 1.2 Logic • Basic Truth Table • Example: • Write the truth table for the following expression:

  8. 1.3 Sets • A set is any well-defined collection of elements • Properties of sets: unordered collection, no duplicate elements • Empty set or null set: set with no elements • If S is contained in T, then S is a subset of T

  9. 1.3 Sets • The intersection of two sets S and T consists of all elements that are both in S and T. • The union of the sets S and T contains all elements that are in either S or T (or both S and T). • The Cartesian product of two sets S and T is the set of all ordered pairs where one element is from S and the other from T

  10. 1.4 Quantifiers • Definition: • Universal quantificationof P(x): P(x) is true for all values of x • Existential quantificationof P(x): There exists an x for which P(x) is true • Quantifier Negation Rules:

  11. 1.4 Quantifiers • Example: • Example:

  12. 1.4 Quantifiers • Example: Negate the following expression

  13. 1.5 Proofs • Methods: • Direct Proof Method • Contrapositive Method • Contradiction Method • “Iff” Method • Other Methods

  14. 1.5 Proofs • Example: • Example:

  15. 1.6 Counterexamples • A conjecture is a result that is thought to be true • A counterexample to a conjecture is an instance where the conclusion to a conditional statement is false • Example:

  16. 4.1 & 4.2 Induction and Recursion • A Mathematical proof procedure used to prove a given statement is true for all natural numbers • Mathematical Induction: • Strong Induction: • How do we know that induction works?

  17. 4.1 & 4.2 Induction and Recursion • A recursive routine is a routine that calls itself (i.e. any given term beyond the base case(s) depend(s) on terms defined in a prior iteration)

  18. How to attack an Induction Problem • First, figure out what we need to prove. • Assign this entity (equation, inequality, or statement) a name, such as P(n). • Figure out if you need Mathematical Induction or Strong Induction • Tip : SI works in all cases MI works • Tip : If the problem is a recursion problem, use SI

  19. 4.1 & 4.2 Induction and Recursion • Example: • Example:

  20. 4.1 & 4.2 Induction and Recursion • Example:

  21. 4.3 Binomial Theorem • Definition: • Key Proposition:

  22. 4.3 Binomial Theorem • Definition: • Example:

  23. 2.1 Division Algorithm • Definition: • Algorithm:

  24. 2.1 Division Algorithm • Example:

  25. 2.2 Euclidean Algorithm • Definition: • Proposition:

  26. 2.2 Euclidean Algorithm • Algorithm:

  27. 2.2 Euclidean Algorithm • GCD Characterization Theorem: • Example:

  28. 2.2 Euclidean Algorithm • Extended Euclidean Algorithm:

  29. 2.2 Euclidean Algorithm • Example: • Propositions:

  30. 2.2 Euclidean Algorithm • Example:

  31. Theorems and Proofs • Theorems to be Memorized • 1.45 Quantifier Negation Rules • 4.13 Principle of Mathematical Induction • 4.18 Principle of Strong Induction • 4.34 Binomial Theorem • 2.12 Division Algorithm • 2.22 Euclidean Algorithm

  32. Theorems and Proofs • Proofs to be Memorized • 4.32 Proposition • 4.34 Binomial Theorem • 2.11 (i) and (ii) Proposition • 2.21 Proposition • 2.27 Proposition • 2.28 Proposition

  33. Thank You! Questions?

  34. Tips • Get a good night’s sleep (unless you’re really really screwed). • Try not to get stuck on one question. • Don’t worry if you don’t get all questions. • Writing down even one line is better than leaving an answer blank. • Your performance in the exam is a reflection of your professor’s teaching. Don’t be nervous. The professor is nervous.

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