PPT-Copyright © Peter Cappello 2011

Predicates amp 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. Selected Exercises. Goal. Explain & illustrate proof construction of . a variety of theorems using strong induction. Copyright © Peter Cappello. 2. Strong Induction. Domain of discussion is the . 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. Use & Misuse. Copyright © Peter Cappello 2011. 2. The Product Rule. If . a procedure has . A 1. st. stage with . s. 1. outcomes . A 2. nd. stage with . s. 2. outcomes. and. . the composite outcomes are . 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. .. Cappello. Mathematical Induction. Goals. . Explain & illustrate construction of . proofs of a variety of theorems using mathematical induction.. Copyright © Peter . Cappello. Motivation. Mathematics uses 2 kinds of arguments:. 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. 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. Goal: . Show . how . propositional equivalences . are established . & introduce . the most . important such . equivalences.. Copyright © Peter . Cappello. 2. Equivalence. Name. p .  T . ≡ p; p . Selected Exercises. Goal. Explain & illustrate proof construction of . a variety of theorems using strong induction. Copyright © Peter Cappello. 2. Strong Induction. Domain of discussion is the . 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. Selected Exercises. Goals. . Introduce . big-O . & big. -. Omega. S. how . how . to estimate . the size of functions using this notation.. Copyright © Peter . Cappello. 2. Preface. You may use . 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?.