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L08_Quantifiers

# L08_Quantifiers - COMP170 Discrete Mathematical Tools for...

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1-1 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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2-1 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
3-1 Variables and Universes

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3-2 Consider the statement: Variables and Universes (*) m 2 > m Is (*) True or False ?
3-3 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

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3-4 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.
4-1 For which values of m is (*) True and for which values is it False ? Again consider the statement: (*) m 2 > m

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4-2 For which values of m is (*) True and for which values is it False ? Again consider the statement: (*) m 2 > m This statement is also ill-defined! The answer depends upon which universe we assume
4-3 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 . 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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4-4 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 Again consider the statement: (*) m 2 > m This statement is also ill-defined! The answer depends upon which universe we assume
4-5 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!

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L08_Quantifiers - COMP170 Discrete Mathematical Tools for...

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