Lecture_3 - 1 MA1100 Lecture 3 Elementary Logic •...

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Unformatted text preview: 1 MA1100 Lecture 3 Elementary Logic • Negation • Conditional Statements • Conjunction and Disjunction • Contrapositive and Converse • Biconditional Statements MA1100 Lecture 3 2 Negation Examples It is raining x is an odd integer The negation of P is given by Ÿ P Say: “not P” The negation of a statement (or predicate) is one that has the exact opposite meaning. negation negation Notation Let P represent a statement (or predicate). If P is true , then Ÿ P is false , and vice versa MA1100 Lecture 3 3 Negation More examples P(x): x = 2 Q(x): x > 2 S: All students hand in homework T: Some students hand in homework MA1100 Lecture 3 4 Negation with quantifier S: For each real number x, x 2 > 0 Example ( " x) [P(x)] Negation: ( $ x) [ Ÿ P(x)] T: All even numbers are divisible by 4 MA1100 Lecture 3 5 Negation with quantifier S: There exists an integer x such that x 2 = 2 Example ( $ x) [P(x)] Negation: ( " x) [ Ÿ P(x)] T: Some even numbers are divisible by 4 MA1100 Lecture 3 6 Negation with two quantifiers To negate two quantifiers , just apply the negation to each quantifier in turn . Ÿ ( " x )( $ y )[P(x, y)] ª ( $ x )( " y ) Ÿ [P(x, y)] Ÿ ( $ x )( " y )[P(x, y)] ª ( " x )( $ y ) Ÿ [P(x, y)] Ÿ ( " x )( " y )[P(x, y)] ª ( $ x )( $ y ) Ÿ [P(x, y)] Ÿ ( $ x )( $ y )[P(x, y)] ª ( " x )( " y ) Ÿ [P(x, y)] ª ( $ x ) Ÿ ( $ y )[P(x, y)] MA1100 Lecture 3 7 Negation with two quantifiers Write down the negation of the following double quantified statements together with their symbolic forms. Exercise 1. For all integers x and y, x + y = 0. 2. There are some integers x and y such that x < y. 3. For every real number y, there exists a real number x such that y = x 2 . 4. There exists an integer x such that for every integer y, xy = 0. MA1100 Lecture 3 8 Conditional Statements A conditional statement is a statement that can be written in the form ‘If S then T’ where S and T are statements or predicates. S is called the hypothesis T is called the conclusion Example (daily life) If you don’t hand in your homework , then you will get 0 mark ‘ you don’t hand in your homework , ’ is the hypothesis ‘ you will get 0 mark ’ is the conclusion Notation S Ø T say: “ S implies T ” MA1100 Lecture 3 9 Conditional with predicates Example (Mathematical) If x is an odd integer , then x is a prime number P(x) and Q(x) are predicates This is actually a universally quantified statement...
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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_3 - 1 MA1100 Lecture 3 Elementary Logic •...

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