MATH121 Section 1 - Laws of Logic - MATH121 Discrete...

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1 MATH121 Discrete Mathematics Bachelor Of Computer Science
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2 Topics Logic Predicate Logic Methods of Proof Set Theory Relations & Functions The Natural Numbers Mathematical Induction The Integers Elementary Number Theory Congruence Arithmetic Graphs and Trees
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3 Why This Course? Propositional logic – digital circuit design Sets/relations - databases (Oracle, MS Access, etc.) Predicate logic - Artificial Intelligence, compilers Proofs - Artificial Intelligence, compilers, theoretical physics/chemistry Congruence arithmetic - Cryptography Graph Theory – network analysis
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4 Logic Section 1
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5 OBJECTIVES Understand statements and compound statements Understand the use of logical connectives to form compound statements Understand the use of the truth table to evaluate the truth values of compound statements Understand what is tautology, contradiction and contingent. Understand the use of logical equivalences to simplify compound statements
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6 What is Logic? Mathematical logic is a tool for dealing with formal reasoning. Numerous application in design of computer circuits, construction of computer programs and verification of correctness of programs.
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7 What is Logic? You are familiar with using numbers in arithmetic and symbols in algebra. You are also familiar with the ‘rules’ of arithmetic and alegbra. Examples: In a similar way, Logic deals with statements or sentences by defining symbols and establishing ‘rules’. (3 + 4) + 6 = 3 + (4 + 6) 3 x – 5 x = (3 – 5) x = 3 + 10 = -2 x = 13
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8 What is Logic? Roughly speaking, in arithmetic an operation is a rule for producing new numbers from a pair of given numbers, like addition (+) or multiplication ( ). In Logic, we form new statements by combining short statements using connectives , like words and , or . Examples: This room is hot and I am tired. MATH122 lectures are fun or I am dreaming.
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9 Statement A statement or proposition is a declarative sentence, that is, a sentence that is either true or false, but not both.
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10 Statement: Examples True statements: 2 + 2 = 4 The sun rises in the east. False statements: 2 + 2 = 7 There are twenty planets in the Solar system.
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11 Non-Statement: Examples Non-statements are questions, commands, exclamations, or sentences with undefined words such as: Study logic. x + y > 0 Do you speak French? Do your homework now. Good! Read this carefully.
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12 Exercises: Which of the following are statements? (i) 2 + 3 = 5 Statement, True (ii) It is raining outside. Statement (iii) 2 + 3 = 6 Statement, False (iv) Is it raining? Non-statement (v) Go away! Non-statement (vi) There exists an even prime number. Statement, True (vii) There are six people here. Statement (viii) Seven is? Non-statement (ix) For some real number x , x < 2. Statement
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13 Exercises: Which of the following are statements?
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