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f2006lecNotesWeek03 - 8 8.1 Predicates and Quantifiers Text...

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8 Predicates and Quantifiers 8.1 Text references we are omitting § 1.4 and § 1.5 in Epp some material in there that may prove useful in other courses gates and circuits — computer organization courses representation of numbers in different bases — possibly APS105 she looks in detail at the representation of integers of size byte we are moving on to Chapter 2 in that chapter, we will be omitting “Tarski’s World” 8.2 Predicates recall that a statement is a sentence that is either true or false (but not both) recall that a sentence such as “ x is less than 10” is not a statement why? — because truth value depends on x we call such a sentence a predicate area of logic dealing with predicates is called predicate calculus we can write predicates symbolically examples we could write “ x is less than 10” as L ( x ) we could write “ x is greater than y ” as G ( x, y ) we could write “ x is a student at U of T” as S ( x ) we can convert a predicate to a statement by assigning a value to the variable(s) examples using preceding predicates e.g. L (12) is a false statement 8.3 Truth sets of predicates in dealing with variables, there is always some domain of discourse e.g. Z , R , ECE students, human beings, etc. the truth set of a predicate is the set of elements in the domain for which the predicate is true as an example, if P ( x ) is the predicate “ x 2 is less than or equal to 25” and the domain is Z + then we could write the truth set of P ( x ) as { x Z + | P ( x ) } here, the value of the truth set is { 1 , 2 , 3 , 4 , 5 } 8.4 The universal quantifier, another way to obtain a statement from a predicate is to use a quantifier suppose domain of discourse is S = { 1 , 2 , 3 } consider the predicate P ( x ): “ x 2 < 10” if we substitute any value from the domain, we always get a true statement i.e. P (1) P (2) P (3) is true we can write this symbolically as: x P ( x ) note that this is a statement , not a predicate more generally, given a domain D , if a predicate is true for every value in D , we write x D, P ( x ) or x D ( P ( x )) 17
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we read this as “For all x in D , P ( x ) is true” sometimes written using conditional x , if x D then P ( x ) or x ( x D P ( x )) this is known as a universal conditional statement if domain of discourse is understood, we may simply write x ( P ( x )) 8.5 The existential quantifier, sometimes we want to assert that a predicate is true (at least) some of the time e.g. at least one person in the room owns a dog let domain of discourse be R , people in this room let D ( x ) be “ x owns a dog” we can write the statement symbolically in the form x R, D (
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