# Lecture_4 - MA1100 Lecture 4 Elementary Logic Logical...

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1 MA1100 Lecture 4 Elementary Logic Logical Equivalence Valid arguments

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MA1100 Lecture 4 2 Logical Equivalence Two statements that involve logical operators are said to be logically equivalent to each other if they have the same truth table. P Q P Ø Q Ÿ P Ÿ Q Ÿ Q ØŸ P TT T T F T T TF F F F T FT T T F T T F FF T We say Ÿ Q P is logically equivalent to P Ø Q We write Ÿ Q P ª P Ø Q
MA1100 Lecture 4 3 Negation of conjunction Is Ÿ (P Q) ªŸ P ⁄Ÿ Q ? Example (daily life) John is tall and thin Negation: John is not tall and thin P Q Ÿ (P Q) n is an even number and a prime Negation: P Q Ÿ (P Q) Example (mathematical)

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MA1100 Lecture 4 4 Negation of disjunction P Q P Q Ÿ (P Q) Ÿ P ⁄Ÿ Q T T T F Ÿ P Ÿ Q Ÿ P ¤Ÿ Q TT F F T T TF F T F FT T FF Is Ÿ (P Q) ªŸ P Q ?
MA1100 Lecture 4 5 De Morgan’s Laws Ÿ (P ¤ Q) ªŸ P ⁄Ÿ Q Ÿ (P Q) P ¤Ÿ Q Example P Q : x is an even number and a prime Ÿ (P Q) : It is not (x is an even number and x is a prime) x is not an even number OR x is not a prime Ÿ P Q

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MA1100 Lecture 4 6 Conjunction with disjunction Is (P ¤ Q) R logically equivalent to P ¤ (Q R) ? Is (P Q) ¤ R logically equivalent to P (Q ¤ R) ? How to construct the truth table? It is clear that (P ¤ Q) ¤ R ª P ¤ (Q ¤ R) and (P Q) R ª P (Q R)
MA1100 Lecture 4 7 Constructing Truth Tables When a statement is made up of more than two simple statements , we need more rows in the truth table to list all combinations of truth values among the simple statements. Example (P Q) ¤ R 8 rows If there are n simple statements in a compound statement, there will be 2 n combinations of truth values.

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MA1100 Lecture 4 8 Constructing Truth Tables P Q R P Q (P Q) ¤ R Example (P Q) ¤ R 8 rows T T T T F F F F T T F F T T F F T F T F T F T F
MA1100 Lecture 4 9 Distributive Laws P ¤ (Q R) ª (P ¤ Q) (P ¤ R) P (Q ¤ R) ª (P Q) ¤ (P R) Example

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## This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.

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Lecture_4 - MA1100 Lecture 4 Elementary Logic Logical...

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