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L08_Quantifiers_print

L08_Quantifiers_print - COMP170 Discrete Mathematical Tools...

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1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 3.2, pp. 104-114 Quantifiers Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen
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2 3.2 Variables and Quantifiers Variables and Universes Quantifiers Standard Notation for Quantification Statements about Variables Proving Quantified Statements True or False Negation of Quantified Statements Implicit Quantification
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3 Consider the statement: Variables and Universes (*) m 2 > m Is (*) True or False ? This is an ill-posed question! For some values of m , e.g., m = 2 , (*) is True For other values of m , e.g., m = 1 / 2 , (*) is False In statements such as m 2 > m , variable m is not constrained . Unconstrained variables are called free variables . Each possible value of a free variable gives a new statement. The Truth or Falsehood of this new statement, is determined by substituting in the new value for the variable.
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4 For which values of m is (*) True and for which values is it False ? For the universe of non-negative integers , the statement is True for every value of m except m = 0 , 1 . For the universe of real numbers , the statement is True for every value of m except for 0 m 1 Two main points: Clearly state the universe A statement about a variable can be True for some values of a variable and False for others. Again consider the statement: (*) m 2 > m This statement is also ill-defined! The answer depends upon which universe we assume
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5 3.2 Variables and Quantifiers Variables and Universes Quantifiers Standard Notation for Quantification Statements about Variables Proving Quantified Statements True or False Negation of Quantified Statements Implicit Quantification
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6 Quantifiers The statement While m 2 > m is True for values such as m = - 3 or m = 9 it is False for m = 0 or m = 1 . Thus, it is not True that m 2 > m for every integer m , so (*) is False (*) For every integer m , m 2 > m is False .
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7 Quantifiers The statement A phrase like for every integer m that converts a symbolic statement about potentially any member of our universe into a statement about the universe is called a quantifier A quantifier that asserts a statement about a variable is true for every value of the variable in its universe , is called a universal quantifier .
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