PPT-Copyright © Peter

Cappello Mathematical Induction Goals Explain amp illustrate construction of proofs of a variety of theorems using mathematical induction Copyright Peter Cappello Motivation Mathematics uses 2 kinds of arguments. Structural Induction:. Selected Exercises. Copyright © Peter Cappello. 2. Exercise 10. Give a recursive definition of . S. m. (. n . ). , the sum of integer . m. . + nonnegative . integer . n. .. Selected Exercises. Copyright © Peter Cappello. 2. Exercise 30. Devise a recursive . algorithm. to find the . n. th. term of the sequence defined by: . a. 0. = 1, . a. 1. = 2. a. n. = . a. n-1. Predicates & Quantifiers. Copyright © Peter Cappello 2011. The Limits of Propositional Logic. Consider the argument. . All. computer science courses are easy.. CS 40 is a computer science course.. Goal: . Show . how . propositional equivalences . are established . & introduce . the most . important such . equivalences.. Copyright © Peter . Cappello. 2. Equivalence. Name. p .  T . ≡ p; p . © Peter . Cappello. Propositional Logic. Copyright . © Peter . Cappello. Sentence Restrictions. Building more precise tools from less precise tools. Precise use of natural language is . difficult. .. Permutations & Combinations:. Selected Exercises. Copyright © Peter Cappello. 2. Exercise 10 (a). A croissant shop has . 6. kinds of croissants: plain, cherry, chocolate, almond, apple, & broccoli.. Selected Exercises. Goal:. Introduce fundamental . number . theory concepts: . T. he . division . algorithm. Congruences. Rules of modular arithmetic. Copyright © Peter . Cappello. 2. Exercise 10. Selected Exercises. Copyright © Peter Cappello. 2. Exercise 30. Devise a recursive . algorithm. to find the . n. th. term of the sequence defined by: . a. 0. = 1, . a. 1. = 2. a. n. = . a. n-1. Goal. : . Introduce predicate logic, . including existential . & universal quantification. Introduce translation between English sentences & logical expressions.. Copyright © Peter . Cappello. Selected Exercises. Goal: . Introduce . computational complexity analysis.. Copyright © Peter Cappello. 2. 2. Exercise 10. How much time does an algorithm take for a problem of size . n. ,. if it uses . Goal: . Show . how . propositional equivalences . are established . & introduce . the most . important such . equivalences.. Copyright © Peter . Cappello. 2. Equivalence. Name. p .  T . ≡ p; p . Structural Induction:. Selected Exercises. Copyright © Peter Cappello. 2. Exercise 10. Give a recursive definition of . S. m. (. n . ). , the sum of integer . m. + nonnegative integer . n. .. Copyright © Peter Cappello. Selected Exercises. Partial Order. Let R be a relation on A.. R is a . partial order . when it is:. Reflexive. Antisymmetric. Transitive.. Copyright © Peter Cappello. 2. Copyright © Peter . Cappello. Copyright © Peter Cappello. 2. Sum Rule Example. There are . 3. sizes of pink shirts & . 7. sizes of blue shirts.. How many . types of shirts. are there, if a . shirt type. is a shirt of a particular color in a particular size?.