Discrete Mathematics DCIT 23 Quick Overview Discrete Math

Discrete Mathematics DCIT 23

Quick Overview Discrete Math is essentially that branch of mathematics that does not depend on limits; in this sense, it is the anti-thesis of Calculus. As computers are discrete object operating one jumpy, discontinuous step at a time, Discrete Math is the right framework for describing precisely Computer Science/Information Technology concepts. Lecture 1 2

Quick Overview The conceptual center of computer science/information technology is the ALGORITHM. Lecture 1 3

Quick Overview Discrete Math helps provide… …the machinery necessary for creating sophisticated algorithms …the tools for analyzing their efficiency …the means of proving their validity Lecture 1 4

What is Discrete Mathematics? • What it isn’t: continuous • Discrete: consisting of distinct or unconnected elements • Countably Infinite • Definition of Discrete Mathematics – Discrete Mathematics is a collection of mathematical topics that examine and use finite or countably infinite mathematical objects. Lecture 1 5

Introduction to Logic Lecture 1 6

Intro We intuitively know that Truth and Falsehood are opposites. That statements describe the world and can be true/false. That the world is made up of objects and that objects can be organized to form collections. The foundations of logic mimic our intuitions by setting down constructs that behave analogously. Lecture 1 7

A Taste of Logic puzzles (1)Knights and Knaves Knights: always tell the truth Knaves: always lie Ø You encounter two people A and B. A says: "B is a knight" B says: "The two of us are opposite types" Q: What are A and B? Lecture 1 8

Why Study Logic? • Why do we need to think logically? Isn’t it better perhaps, to be spontaneous, intuitive and perceptive? – Reasoning is • drawing out conclusion on the basis of certain evidence, • infer one thing from another or • to figure out consequences of a certain course of action.

Why Study Logic? • to recognize and use certain very common forms of correct logical inference and avoid certain common logical errors. • To increase our ability to construct extended chains of reasoning and to deal with more complex problems: – Consider a number of options, consequences of each of these options and consequences of the consequences • To not just how to reason correctly, but also why certain forms of inference are correct and others incorrect.

Logic • Is concerned with verbal expression of reasoning (argument), whether arguments are good or bad, correct or incorrect • An argument - a set of sentences consisting of one or more premises, which contain the evidence, and a conclusion, which is supposed to follow from the premises.

Example of an argument John will not get an A on this exam unless he studied hard for it. John did not study hard for this exam. Therefore, John will not get an A on this exam.

Correctness of argument • Depends on the connection between premises and conclusion, and not on whether the premises are true or not. • It is generally the task of other disciplines to assess the truth or falsity of particular statements

A valid deductive argument • Has the strongest conceivable kind of logical relationship with all premises that are true, then the conclusion established by the argument is true as well.

Induction vs. Deduction • Distinction lies in the intended strength of the connection between premises and conclusion. • In a deductive argument, the premises are intended to provide total support for the conclusion • In an inductive argument, the premises are only supposed to provide some degree of support for the conclusion.

Inductive or Deductive? • John gets A’s on 95% of the exams for which he studies, and that he did study for this exam, therefore, he will probably get an A on this exam. • 99. 9% of commercial airline flights are completed without incident, we may correctly infer that the next airplane we take will almost certainly arrive safely.

Induction or Deduction? John cannot get into law school unless he scores well on the LSAT exam John did not score well on the exam Therefore, John cannot get into law school.

Induction vs. Deduction • Deduction: either the premises do provide absolute support for the conclusion, in which case the argument is valid, or they do not, in which case it is invalid. • Induction: the “goodness” of an argument is a matter of degree.

Scope and limitation • This course is concerned exclusively with deductive logic because: – It is so clear-cut – It is easier and better suited to a foundation to math course – At present, there is no generally accepted system of inductive logic

Valid vs. Invalid arguments • A valid deductive argument is one in which the truth of the premises absolutely guarantees the truth of the conclusion. • An invalid argument, by contrast, is one in which it is possible for the premises all to be true but the conclusion false.

Form 1. All horses are green. All green things are plastic. ------------------- All horses are plastic. 2. If Noynoy is the president, then Binay is the vice-president. ------------------- Noynoy is the president. a valid argument an invalid argument this is all rather abstract, isn’t it?

Form and Validity • It is the form of an argument that determines its logical validity. Thus, • Symbolic logic turns out to be primarily the study of the abstract forms and structures used in argumentation.

Form and Validity • Consider the ff, notion of two arguments having the same form: 1. Either F. Marcos was elected 2. Jose is either at the movies or at home. in 1986 or Cory Aquino was Jose is not at home. elected in 1986. -------------------F. Marcos was not elected in Jose is at the movies. 1986. -------------------Either p or q Cory Aquino was elected in 1986. not q What is the pattern? ------------- p

Sentential Logic • Complete simple sentences are taken as unbroken units and are not analyzed into their component parts. • The form of a compound sentence or argument is then determined by how these simple sentences are combined with certain logical words, such as “and”, “or”, and “not”.

Form vs. Instances • The form is the general pattern or structure, which abstracts from all specific subject matter • Instances are the particular meaningful examples that exhibit that form.

Instance • In sentential logic, it is obtained from a form by substituting meaningful sentences (consistently) for the p’s and q’s; • In predicate logic, instances are obtained by substituting class terms for the capital letters A, B, and C.

Advantages of using symbols • Easier to manipulate • They provide economical shorthand • Allow us to see at a glance the overall structure of a sentence

Symbolic system of sentential logic • Single letter to stand for simple sentences, such as “Juan enrolled in Geometry” • Special symbols for the five logical words “and”, “or”, “not”, “if-then”, and “if-and-onlyif” • Grouping symbols such as parentheses • No attempt of analysis of simple sentences

Applications of Logic • It is the foundation for expressing formal proofs in all branches of mathematics. • In computer science/information technology: ØCircuit design ØProgram constructing ØProgram verification Lecture 1 29