Lecture_2 - MA1100 Lecture 2 Elementary Logic Statements...

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1 MA1100 Lecture 2 Elementary Logic Statements and Predicates Quantified Statements Negation

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MA1100 Lecture 2 2 Mathematical Language ± Not exactly the same as English ± A dialect of English ± Expressing mathematical ideas precisely ± The grammar is Logic Sentences Statements Predicates Others
MA1100 Lecture 2 3 Statements A statement is a sentence that is either true or false ( but not both ). Examples (daily life) 1) MA1100 lectures are on Tuesday and Friday 2) MA1100’s lecturer is a woman 3) MA1100 is an easy module 4) Is MA1100 lecture conducted in LT27? 5)I always lie

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MA1100 Lecture 2 4 Statements A statement is a sentence that is either true or false ( but not both ). Examples (Mathematical) 1) 3 + 4 = 8 2) 2x + 5y = 7 3) 2x + 5y = 7 is a linear equation in x and y. 4) There are integers x and y such that 2x + 5y = 7 5) For all integers x and y, 2x + 5y = 7
MA1100 Lecture 2 5 Predicates A predicate is a (mathematical) sentence that involves variables . With x = 1, y = 1 , 2x + 5y = 7 is true. With x = 3, y = 2 , 2x + 5y = 7 is false. When the variables are substituted with specific values , a predicate becomes a statement . 2x + 5y = 7 is an example of a predicate Example

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MA1100 Lecture 2 6 Predicates 1) He thinks MA1100 is an easy module. 2) She says he is cute. He ” is a variable Substitute “ He ”by ±“ The lecturer The lecturer thinks MA1100 is an easy module.
MA1100 Lecture 2 7 Predicates 1) n is an odd number 2) p is a prime number 3) xy = yx 4) sin 2 θ + cos 2 θ = 1 More examples (Mathematical) When it is clear from the context, we may omit the quantifiers and regard a predicate as a statement.

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MA1100 Lecture 2 8 Using the Equal sign = 1) 3 + 4 = 5 + 2 2) 2x + 5y = 7 3) sin 2 θ + cos 2 θ = 1 4) f(x) = x 2 Examples equality equation identity assignment
MA1100 Lecture 2 9 Symbolic Representation We denote a statement by capital letters , usually P, Q, R P : 2 is an even number Example means P represents the statement ‘ 2 is an even number Q : 7 is a negative number means Q represents the statement ‘ 7 is a negative number

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MA1100 Lecture 2 10 Symbolic Representation P(n) : n is an odd number Q(x,y) : 2x + 5y = 7 Example We denote a predicate by capital letters with the variables involved, such as P(n), Q(x,y), P(2) : 2 is an odd number P(3) : 3 is an odd number
MA1100 Lecture 2 11 Quantified Statements There are integers x and y : existential quantifier

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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_2 - MA1100 Lecture 2 Elementary Logic Statements...

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