MAT2345 Discrete Math The Course Propositional Logic Dr. Van - PowerPoint PPT Presentation

MAT2345 Discrete Math Dr. Van Cleave Guidelines MAT2345 Discrete Math The Course Propositional Logic Dr. Van Cleave Propositional Equivalences Predicates & Quantifiers Fall 2013 Nested Quantifiers Dr. Van Cleave MAT2345 Discrete Math

General Guidelines MAT2345 Discrete Math Dr. Van Syllabus Cleave Guidelines Schedule (note exam dates) The Course Propositional Logic Homework, Worksheets, Quizzes, Propositional and possibly Programs & Reports Equivalences Predicates & Quantifiers Academic Integrity Guidelines — Do Your Own Work Nested Quantifiers Course Web Site: www.eiu.edu/~mathcs Dr. Van Cleave MAT2345 Discrete Math

Course Overview MAT2345 Discrete An introduction to the mathematical foundations Math needed by computer scientists. Dr. Van Cleave Logic & proof techniques Guidelines The Course Sets, functions Propositional Logic Algorithms – developing and analyzing Propositional Equivalences Recursion & induction proofs Predicates & Quantifiers Recurrence relations Nested Quantifiers If time permits: Boolean algebra, logic gates, circuits Modeling computation Dr. Van Cleave MAT2345 Discrete Math

Course Themes MAT2345 Discrete Math Mathematical Reasoning – proofs, esp by induction Dr. Van Cleave Mathematical Analysis – comparison of algorithms, Guidelines function growth rates The Course Propositional Logic Discrete Structures – abstract math structures, Propositional the relationship between discrete and abstract Equivalences structures Predicates & Quantifiers Nested Algorithmic Thinking – algorithmic paradigms Quantifiers Applications and Modeling – can we predict behavior? Dr. Van Cleave MAT2345 Discrete Math

Student Responsibilities — Week 1 MAT2345 Discrete Math Dr. Van Cleave Guidelines Reading : Textbook, Sections 1.1 – 1.4 The Course Propositional Logic Assignments : See Homework Assignments Handout Propositional Equivalences Attendance : Strongly Encouraged Predicates & Quantifiers Nested Quantifiers Dr. Van Cleave MAT2345 Discrete Math

Week 1 Overview MAT2345 Discrete Math Dr. Van Cleave 1.1 Propositional Logic Guidelines The Course 1.2 Propositional Equivalences Propositional Logic Propositional 1.3 Predicates and Quantifiers Equivalences Predicates & Quantifiers 1.4 Nested Quantifiers Nested Quantifiers Dr. Van Cleave MAT2345 Discrete Math

Section 1.1 Propositional Logic MAT2345 Discrete Math The rules of logic are used to distinguish between valid Dr. Van and invalid mathematical arguments. Cleave Guidelines Logic rules have many applications in computer science. The Course They are used in: Propositional Logic the design of computer circuits Propositional Equivalences the construction of computer programs Predicates & Quantifiers the verification of the correctness of programs Nested Quantifiers as the basis of some Artificial Intelligence programming languages. and many other ways as well Dr. Van Cleave MAT2345 Discrete Math

Propositions MAT2345 Discrete PROPOSITION : a statement that is either true or false, Math but not both. Dr. Van Cleave Examples (which are true ?): Guidelines The zip code for Charleston, IL is 61920. The Course The Jackson Avenue Coffee Shop is located on Jackson Propositional Logic Avenue. Propositional 1 + 4 = 5 Equivalences 1 + 3 = 5 Predicates & The title of our course is Mathemagics. Quantifiers Nested Counterexamples : Quantifiers Where am I? Stop! x + 2 y = 4 Dr. Van Cleave MAT2345 Discrete Math

Vocabulary MAT2345 Variables are generally used to represent propositions: Discrete Math p , q , r , s , . . . Dr. Van Cleave Tautology : a proposition which is always true . Guidelines The Course Contradiction : a proposition which is always false . Propositional Logic Propositional Equivalences Compound Proposition : a new proposition formed from Predicates & existing propositions using logical operators (aka Quantifiers connectives ). Nested Quantifiers Negation : let p be a proposition. The negation of p is the proposition “ It is not the case that p ,” denoted by ¬ p or ∼ p . Dr. Van Cleave MAT2345 Discrete Math

Truth Tables MAT2345 Truth Tables display the relationship between the truth values Discrete Math of propositions. Dr. Van Cleave The truth table for negation : Guidelines The Course ¬ p p Propositional Logic T F Propositional F T Equivalences Predicates & Quantifiers Nested When proposition p is true, its negation is false. Quantifiers When it is false, its negation is true. The negation of “ Today is Monday ” is “ Today is not Monday ” or “ It is not the case that today is Monday ” Dr. Van Cleave MAT2345 Discrete Math

Conjunction MAT2345 Discrete Math Conjunction : the compound proposition p and q , or p ∧ q Dr. Van Cleave which is true when both p and q are true Guidelines and false otherwise. The Course Propositional Logic Let p = Today is Monday , and q = It is raining . Propositional What is the value of each of the following conjunctions? Equivalences Predicates & Quantifiers p ∧ q Nested Quantifiers p ∧ ¬ q Dr. Van Cleave MAT2345 Discrete Math

Disjunction MAT2345 Discrete Math Disjunction : the compound proposition p or q , or p ∨ q Dr. Van which is false when both p and q are false Cleave and true otherwise. Guidelines The Course p ∧ q p ∨ q Propositional p q Logic T T T T Propositional Equivalences T F F T Predicates & Quantifiers F T F T Nested Quantifiers F F F F Dr. Van Cleave MAT2345 Discrete Math

Exclusive Or MAT2345 Exclusive Or : p ⊕ q , the proposition that is true when exactly Discrete Math one of p and q is true , and is false otherwise. Dr. Van Cleave “Fries or baked potato come with your meal” Guidelines “Do the dishes or go to your room” The Course Propositional Logic Propositional p ⊕ q p q Equivalences Predicates & T T F Quantifiers Nested T F T Quantifiers F T T F F F Dr. Van Cleave MAT2345 Discrete Math

Implication Implication : p → q ( if p then q ), the proposition that is MAT2345 Discrete true unless p is true and q is false (i.e., T → F is false ). Math Dr. Van Cleave p is the antecedent or premise Guidelines q is the conclusion or consequence The Course Propositional Logic Implication Propositional Equivalences p → q p q Predicates & Quantifiers T T T Nested Quantifiers T F F F T T F F T Dr. Van Cleave MAT2345 Discrete Math

Implications Related To p → q MAT2345 Discrete Math p → q Dr. Van Direct Cleave Statement Guidelines q → p The Course Converse Propositional ¬ p → ¬ q Logic Inverse Propositional Equivalences ¬ q → ¬ p Contrapositive Predicates & Quantifiers p ↔ q or p iff q Nested Biconditional Quantifiers the proposition which is true when p and q have the same truth values, and false otherwise. Dr. Van Cleave MAT2345 Discrete Math

Example MAT2345 Discrete Math Direct : If today is Monday, then Dr. Van MAT2345 meets today . Cleave Guidelines Converse : If MAT2345 meets today, The Course then today is Monday . Propositional Logic Inverse : If today is not Monday, Propositional Equivalences then MAT2345 does not Predicates & meet today . Quantifiers Nested Quantifiers Contrapositive : If MAT2345 does not meet today, then today is not Monday . Dr. Van Cleave MAT2345 Discrete Math

Implications — aka Conditionals MAT2345 Converse, Inverse, and Contrapositive Discrete Math Dr. Van Direct Statement p → q If p , then q Cleave Converse q → p If q , then p Guidelines Inverse ∼ p →∼ q If not p , then not q The Course Propositional ∼ q →∼ p If not q , then not p Contrapositive Logic Propositional Equivalences Let p = “they stay” and q = “we leave” Predicates & Direct Statement ( p → q , in English): Quantifiers Nested Converse : Quantifiers Inverse : Contrapositive : Dr. Van Cleave MAT2345 Discrete Math

MAT2345 Discrete Math Let p = “I surf the web” and q = “I own a PC” Dr. Van Cleave Direct Statement ( p → q ): Guidelines The Course Propositional Converse : Logic Propositional Equivalences Predicates & Quantifiers Inverse : Nested Quantifiers Contrapositive : Dr. Van Cleave MAT2345 Discrete Math

Equivalent Conditionals MAT2345 Discrete Math Direct Converse Inverse Contrapositive Dr. Van Cleave p → q q → p ∼ p → ∼ q ∼ q → ∼ p Guidelines ∼ p ∨ q p q The Course T T T T Propositional Logic T F F T Propositional Equivalences F T T F Predicates & Quantifiers F F T T Nested Quantifiers � → △ is equivalent to ∼ � ∨ △ ∼ � ∨ △ ≡ � → △ � ∨ △ ≡ ∼ � → △ Dr. Van Cleave MAT2345 Discrete Math

Tricky Question MAT2345 Discrete Math For the expression p ∨ q , write each of the following in symbols: Dr. Van Cleave Direct Statement : Guidelines The Course Propositional Converse : Logic Propositional Equivalences Inverse : Predicates & Quantifiers Nested Quantifiers Contrapositive : Dr. Van Cleave MAT2345 Discrete Math