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# section02 - MAT 300 Spring 2009 Chad Awtrey SCHOOL OF...

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Unformatted text preview: MAT 300 Spring 2009 Chad Awtrey SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES Section 2 — Quantifiers EXERCIES Variable Statements Consider the sentence x 2 5 x + 6 = This sentence has a variable x , therefore it is customary to use function notation when referring to it. Thus we write p ( x ) : x 2 5 x + 6 = For a specified value of x , p ( x ) becomes a statement that is either true or false. For example p ( 2 ) is true p ( 4 ) is false MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 17 Section 2 — Quantifiers EXERCIES Universal Quantifier 8 When a variable is used in an equation or inequality, we assume that the general context for the variable is the set of real numbers, unless we are told otherwise. Within this context, we can remove the ambiguity of p ( x ) by using a quantifier : For every x , x 2 5 x + 6 = is a statement since it is false. In symbols, we write 8 x ; p ( x ) where the universal quantifier 8 is read as either “For all...” “For every...” “For each...” or some similar phrase MAT 300 Awtrey MATHEMATICS AND STATISTICS 3 / 17 Section 2 — Quantifiers EXERCIES Existential Quantifier 9 The sentence There exists an x such that x 2 5 x + 6 = 0. is also a statement since it is true. In symbols we write 9 x s.t. p ( x ) where the existential quantifier 9 is read as either “There exists...” “There is at least one...” or some similar prhase The symbol “s.t.” is just shorthand notation for the phrase “such that”. MAT 300 Awtrey MATHEMATICS AND STATISTICS 4 / 17 Section 2 — Quantifiers EXERCIES Quantifiers Example The statement There exists a number less than 7 can be written as 9 x s.t. x < 7 Sometimes the quantifier is not explicitly written down, as in...
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section02 - MAT 300 Spring 2009 Chad Awtrey SCHOOL OF...

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