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PredicateLogic

# PredicateLogic - CSE 1400 Applied Discrete Mathematics...

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CSE 1400 Applied Discrete Mathematics Predicates Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Predicate Logic 1 Predicate Logic Inference Rules 7 Negation of Quantified Predicate Statements 7 Reasoning about Quantification Order 8 Problems on Predicate Logic 8 Predicate Logic A Boolean statement is either True or False . By contrast, the truth Predicate logic is also called first-order logic. value of a predicate statement depends on the value one or more variables, and functions and relations between them. • A predicate statement can be True for all values of the variables. Logicians call this a universal affirmative . An instance of a universally True predicate statement is “For all natural number, the sum of the first n is n ( n - 1 ) /2.” Call “The sum (of the first n natural numbers)” the subject and label it S . You can think of S as the set of numbers that are sums of first n natural numbers S = { 0, 1, 3, 6, 10, 15, . . . } Call “ n ( n - 1 ) /2” the predicate and label it P . You can think of P as the set of numbers that are of the form n ( n - 1 ) /2 where n is a natural number P = n ( n - 1 ) 2 : n N The predicate statement above can be represented by the Euler diagram

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cse 1400 applied discrete mathematics predicates 2 S P U which shows every s S is a member of P . A universally affirmative predicate statement can be repre- sented by the logic formula ( s )( s S s P ) which states that for all values s that are the sums of the natural numbers (starting at 0 and stopping at some finite value n ) can be computed by the formula n ( n - 1 ) /2 for some natural number n . There are, of course, others ways to express this, for instance, ( s S )( n N )( s = n ( n - 1 ) /2 ) • A predicate statement can be True for no values of the variables. Logicians call this a universal negative . An instance of a uni- versally False predicate statement is “For all natural numbers is the sum of the first n natural numbers is not equal to 7.” This predicate statement can be represented by the Euler diagram S P U which shows no element s S is a member of P = { 7 } and vice versa. A universally negative statement can be written as ( s )( s S s 6∈ P ) which states if s is the sum of natural numbers (starting at 0 and stopping at some finite value n ) then s is not equal to 7. • A predicate statement can be True for at least one list of values for the variables. Logicians call this a particular affirmative . An instance of predicate statement that is True for some values of the variable is
cse 1400 applied discrete mathematics predicates 3 “The sum of the first n natural numbers is 10.” This predicate statement can be represented by the Euler diagram S P U which shows there is a non-empty intersection between S and P = { 10 } .

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