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. Copyright © Peter Cappello 2011. 2. Exercise 10. Which relations in Exercise 4 are irreflexive?. A relation is . irreflexive. .  . a .  . A (a, a) . . . R.. Ex. 4 relations on the set of all people:. Selected Exercises. Goal:. Introduce fundamental . number . theory concepts: . T. he . division . algorithm. Congruences. Rules of modular arithmetic. Copyright © Peter . Cappello. 2. Exercise 10. 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. 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 . 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 . Votkinsk. in 1840. . P. eter wrote his music during a time known as the Romanic period. He used his great imagination create beautiful music that was sometimes very happy and sometimes very sad.. Peter Tchaikovsky first became interested in music when he was four or five years old. One day Peter’s father brought home a large machine that played music called an . Selected Exercises. Copyright © Peter Cappello 2011. 2. Exercise 10. Which relations in Exercise 4 are irreflexive?. A relation is . irreflexive. .  . a .  . A (a, a) . . . R.. Ex. 4 relations on the set of all people:. 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 . Goals. : . Explain . how to work with nested . quantifiers. S. how that . the order . of quantification . matters. . Work . with . logical . expressions involving multiple . quantifiers.. Copyright © . 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?.