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Intro to Analysis in-class_Part_5

Intro to Analysis in-class_Part_5 - Chapter 1 Preliminaries...

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Chapter 1 Preliminaries 1.1 Quantifiers 1 Sentential logic/propositional calculus: A, B are absolute statements about the state of affairs. Any expression like A is (globally) true or false. How to express more delicate ideas: relations between specific objects/individuals, etc? Predicate logic (aka 1st order logic): A ( x ) , B ( x ) are statements about a variable x . It may be that A ( n ) is true but A ( m ) is false! So how to express when something is always true or sometimes true or never true? Use quantifiers. Definition 1.1.1 (Universal quantifier) . If A ( x ) is true for every possible value of x (under discussion), we say x, A ( x ). Example 1.1.1. x 2 0 , x R . TRUE x, a < b = a 2 < b 2 . FALSE x 0 , a < b = a 2 < b 2 . TRUE x (0 , 1) , n, x n < x . TRUE (assume n N ). 1 May 2, 2007
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16 Math 413 Preliminaries Definition 1.1.2 (Existential quantifier) . If A ( x ) is true for at least one allowable value of x , we say x, A ( x ), or x such that A ( x ).
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