L08_Quantifiers

L08_Quantifiers - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 3.2, pp. 104-114QuantifiersVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-13.2 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 Quantification3-1Variables and Universes3-2Consider the statement:Variables and Universes(*)m2> mIs(*)TrueorFalse?3-3Consider the statement:Variables and Universes(*)m2> mIs(*)TrueorFalse?This is an ill-posed question!For some values ofm, e.g.,m= 2,(*)isTrueFor other values ofm, e.g.,m= 1/2,(*)isFalse3-4Consider the statement:Variables and Universes(*)m2> mIs(*)TrueorFalse?This is an ill-posed question!For some values ofm, e.g.,m= 2,(*)isTrueFor other values ofm, e.g.,m= 1/2,(*)isFalseIn statements such asm2> m, variablemisnot constrained.Unconstrained variables are calledfree variables.Each possible value of a free variable gives anewstatement.TheTruthorFalsehoodof this new statement, is determinedby substituting in the new value for the variable.4-1•For which values ofmis(*)Trueand for which values is itFalse?Again consider the statement:(*)m2> m4-2•For which values ofmis(*)Trueand for which values is itFalse?Again consider the statement:(*)m2> m•This statement is also ill-defined!The answer depends upon whichuniversewe assume4-3•For which values ofmis(*)Trueand for which values is itFalse?•For the universe ofnon-negative integers, the statementisTruefor every value ofmexceptm= 0,1.Again consider the statement:(*)m2> m•This statement is also ill-defined!The answer depends upon whichuniversewe assume4-4•For which values ofmis(*)Trueand for which values is itFalse?•For the universe ofnon-negative integers, the statementisTruefor every value ofmexceptm= 0,1.•For the universe ofreal numbers, the statement isTruefor every value ofmexcept for≤m≤1Again consider the statement:(*)m2> m•This statement is also ill-defined!The answer depends upon whichuniversewe assume4-5•For which values ofmis(*)Trueand for which values is itFalse?•For the universe ofnon-negative integers, the statementisTruefor every value ofmexceptm= 0,1.•For the universe ofreal numbers, the statement isTruefor every value ofmexcept for≤m≤1Two main points:•Clearly state the universe•A statement about a variable can beTruefor some valuesof a variable andFalsefor others.Again consider the statement:(*)m2> m•This statement is also ill-defined!The answer depends upon whichuniversewe assume5-13.2 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 Quantification6-1Quantifiers6-2QuantifiersThe statement(*)For every integerm,m2> misFalse.6-3...
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

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L08_Quantifiers - 1-1COMP170Discrete Mathematical Toolsfor...

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