Discrete Struc CECS 228
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This 8 page Class Notes was uploaded by Zackary Cronin on Monday October 5, 2015. The Class Notes belongs to CECS 228 at California State University - Long Beach taught by Todd Ebert in Fall. Since its upload, it has received 20 views. For similar materials see /class/218746/cecs-228-california-state-university-long-beach in Computer Science and Engineering at California State University - Long Beach.
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Date Created: 10/05/15
Lecture 1 Propositional Logic 0 Associated reading Rosen7 Section 11 o Homework Section 117 problems 1 23 odd7 37 42 Ingredients of logic 0 used for expressing rules and rationality within a system ie modeling a system 0 rules tend to be expressed using logical formulae which may be Viewed in two funda me nt al ways semantics refers to the meaning behind a string of symbols 7 syntax refers to the structure of a string of symbols and the rules which govern the creation of strings of symbols 0 rationality within a system tends to depend upon a set of axioms and rules ofinferenee Some different types of logics that are of importance to computer science 0 PropositionalBoolean logic the design of combinatorial electronic circuits pro gram control Predicate logic formal speci cation7 prolog programming language 0 Higherorder logic formal veri cation of hardware 0 Sequentialtemporal logic design of computer memory program concurrency o Intensional logic natural language processing human computer interaction Propositional Logic lntuitively7 a proposition represents a t t t that vuiuut s to either true or false Example 1 Provide some examples of propositional statements Example 2 Provide some examples of non propositions Atomic propositions are types of propositions that represent one single idea 0 the truth or falsehood of an atomic proposition does not depend upon the truth or falsehood of any sub propositions ie it is either true or false in and of itself 0 captial letters P7 Q71 7 will denote atomic propositions Example 3 Provide some examples of atomic propositions compound propositions are propositions that can be expressed using one or more atomic propositions and one or more logical connectives o logical connectives operators are used to build more complex propositional statements NOT a AND OR V EXOR EB lF THEN a EQUIVALENCE lt gt oncnqgtcowgta o lowercase letters 104177 7 will denote compound propositions 0 compound propositions are also called propositional formulae Example 4 Provide some examples of compound propositions Truth tables provide a means for determining the truth or falsehood of a compound propo sition based upon the truth or falsehood of its constituent atoms Example 5 Make truth tables for each of the logical connectives Example 6 Make a truth table for the compound proposition P a Q V R Example 7 Write the formula P lt gt Q R V P using parentheses IfThen statements Propositions of the form p a q are called If then statements and occur quite often both explicitly and implicitly in natural language Here are a few ways which one may encounter an if then statement within the natural language context 0 p impies q High taxes imply high unemployment o if pq this form is used often as a program statement 0 p only if q you will pass this class only if you study 0 p is suf cient for q receiving an A on all exams is suf cient for receiving an A in the class 0 q if p I will visit Tahiti if the air fares seem low 0 q whenever p we will go whenever Joe gets here 0 q is necessary for p studying is necessary for passing this class For the statement p a q o p is often called the hypothesis7 antecedent7 or premise o q is often called the 39 39 or Given the proposition p a Q7 0 q a p is called its converse 0 g a p is called its contrapositive o it is called trivially true or vacuouslly true whenever p is false Example 7 Given an example of a trivially true statement Write out its converse and contrapositive