07a-Predicates-and-Quantifiers

07a-Predicates-and-Quantifiers - Predicates and...

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Predicates and Quantifiers Discrete Mathematics
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Discrete Mathematics – Predicates and Quantifiers 7-2 What Propositional Logic Cannot Do We saw that some declarative sentences are not statements without specifying the value of `indeterminates’ Some valid arguments cannot be expressed with all our machinery of tautologies, equivalences, and rules of inference Every man is mortal. Socrates is a man. Socrates is mortal ``A man has a brother’’ ``If x + 1 > 0, then x > 0’’ `` x + 2 is an even number’’
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Discrete Mathematics – Predicates and Quantifiers 7-3 Open Statements or Predicates Sentences like `x is greater than 3’ or `person x has a brother’ are not true or false unless the variable is assigned some particular value. Sentence `x is greater than 3’ consists of 2 parts. The first part, x, is called the variable or the subject of the sentence. The second part – the predicate, `is greater than 3’ – refers to a property the subject can have. Sentences that have such structure are called open statements or predicates We write P(x) to denote a predicate with variable x
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Discrete Mathematics – Predicates and Quantifiers 7-4 Unary, Binary, and so on `x is greater than 3’ `x is my brother’ `x is a human being’ contain only 1 variable, unary predicates `x is greater than y’ `x is the mother of y’ `car x has colour y’ contain 2 variables, binary predicates `x divides y + z’ `x sits between y and z’ `x is a son of y and z’ P(x) Q(x,y) R(x,y,z) contain 3 variables, ternary predicates
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Discrete Mathematics – Predicates and Quantifiers 7-5 Assigning a Value When a variable is assigned a value, the predicates turns into a statement, whose truth value can be evaluated. P(x) = `x is greater than 3’ P(2) = `2 is greater than 3’ P(4) = `4 is greater than 3’ x=2 x=4 false true Q(x,y) = `car x has colour y’ Q(my car,red) = `my car is red’ Q(my car,grey) = `my car is grey’ x=my car y=red x=my car y=grey false true
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Discrete Mathematics – Predicates and Quantifiers 7-6 Universe We cannot assign a variable of a predicate ANY value.
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07a-Predicates-and-Quantifiers - Predicates and...

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