lecture 1- 5 - DISCRETE MATHEMATICS AND ITS APPLIACTIONS By...

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DISCRETE MATHEMATICS AND ITS APPLIACTIONS By the end of the lecture (i) you should be familiar with the following terms: Proposition Truth value Propositional calculus/propositional logic Negation, conjunction, disjunction Compound proposition Exclusive or, inclusive or Implications Converse, contrapositive, inverse (ii) be able to Translate English sentences into expressions involving propositional variables and logical connectives Section 1.1 Text Text: Discrete Mathematics and its Applications, Kenneth Rosen, Mc Graw Hill, fifth ed., 2003
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Lecture 1 Formal logic The rules of logic give precise meaning to mathematical statements. The rules are used to distinguish between valid and invalid mathematical statements. Logic is the basis of all mathematical reasoning. It has practical application in the design of computing macnines, artificial intelligence, programming languages and many other areas in computer science. Propositions Let us begin with the building blocks of logic – propositions . A proposition is a statement that is either true or false, but not both. Example: The following are propositions: 5 + 5 = 10 true 1 + 6 = 2 false Owen Arthur is the Prime Minister of Barbados true The following are NOT propositions since their truth value cannot be established: r + s = t x + 5 = 7 Letters are used to denote propositions. (eg. p or q ). For example the propositions above could have been written as p: 5 + 5 = 10 q: 1 + 6 = 2 r: Owen Arthur is the Prime Minister of Barbados The truth value of a proposition is true, denoted by T , if it is a true proposition, and false F , if it is a false proposition. The area of logic which deals with propositions is called propositional calculus or propositional logic . Def: Let p be a proposition. The negation of p, denoted by ~p, is the proposition “it is not the case that p” Compound propositions are formed from existing propositions using logical operators called connectives . Some of these connective are the conjunction (AND denoted ), the disjunction (OR denoted ), implication (denoted ),the equivalence (
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Def: Let p and q be propositions. The proposition “p and q” denoted by p q , is the proposition that is true when both p and q are true and is false otherwise. The proposition p q is called the conjunction of p and q. Def: Let p and q be propositions. The proposition “p or q” denoted by p q, is the proposition that is false when both p and q are false and is true otherwise. The proposition p q is called the disjunction of p and q. This is often referred to a the inclusive or . Example: Let p be the proposition “Today is Monday” and q the proposition “It is the first Discrete Math class”. Find the negation of
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lecture 1- 5 - DISCRETE MATHEMATICS AND ITS APPLIACTIONS By...

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