New PDF release: Discrete Mathematics for Computer Science Some Notes

By Gallier J.

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For this, given any set, A, and given a property, P , (specified by a first-order formula) we need to be able to define the subset of A consisting of those elements satisfying P . This subset is denoted by {x ∈ A | P }. Unfortunately, there are problems with this construction. If the formula, P , is somehow a circular definition and refers to the subset that we are trying to define, then some paradoxes may arise! The way out is to place a restriction on the formula used to define our subsets, and this leads to the subset axioms, first formulated by Zermelo.

Iv) The equality predicate, =, is added to our language when we want to deal with equations. (v) First-order variables, t1 , t2 , . . , used to form quantified formulae. The difference between function symbols and predicate symbols is that function symbols are interpreted as functions defined on a structure (for example, addition, +, on N), whereas predicate symbols are interpreted as properties of objects, that is, they take the value true or false. An example is the language of Peano arithmetic, L = {0, S, +, ∗, =}.

7), we know that there is no computer program that will terminate for all input propositions and give an answer in a finite number of steps! So, although the function Prov makes sense as an abstract function, it is not computable. Is this a paradox? No, if we are careful when defining a function not to incorporate in the definition any notion of computability and instead to take a more abstract and, in some some sense naive view of a function as some kind of input/output process given by pairs input value, output value (without worrying about the way the output is “computed” from the input).

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Discrete Mathematics for Computer Science Some Notes by Gallier J.


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