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# Discrete Math - Week 3 CS 2305

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This 2 page Class Notes was uploaded by Aaron Maynard on Tuesday February 2, 2016. The Class Notes belongs to CS 2305 at a university taught by Timothy Farage in Spring 2016. Since its upload, it has received 44 views.

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Date Created: 02/02/16

Discreet Math for Computing Aaron Maynard th Timothy Farage January 26 -28th, 2016 Predicate Logic (First Order Logic) Definition: A predicate is a propositional function. Its value is either true or false. F:R->R where F(x) = x 2 U x *N = {0, 1, 2, 3, …} P(x) = x > 7 Q(x,y) = x + y = 10 S(x): x is smart, U =x{women} Predicates can also be made into propositions using quantifiers. Definition: A formal expression used in asserting that a stated general proposition is true of all of the members of the delineated universe or class. Universal Quantifier: ∀ = fxr all x, for every x; Existential Quantifier: ∃ = xhere exists an x, there exists at least one x; Examples using s(x,y): x+y=10 U + U = Nx= {0,y1, 2, 3, …} ∃ x∃ *ys(x,y)] = There is a natural number x, there is a natural number y, such that x + y = 10. ∀ *∀ *[s(x,y)] = For every natural number x, for every natural number y, x + y = x y 10. ∀ x∃ *xs(x,y)] = For every natural number x, there exists a natural number y, x + y = 10. ∃ x∀ *xs(x,y)] = There is a natural number x, for every natural number y, x + y = 10. Examples using integers: P(x) = x is prime, U + U = N x y ∀ x[P(x) -> ∃ *yP(y) Λ(y>x)] = For every natural number x, if x is prime, then there is a natural number such that y is prime and y > x. Discreet Math for Computing Aaron Maynard th Timothy Farage January 26 -28th, 2016 U x U = ykitty cats} ∀ x For all x ∃ x There exists (at least one) x S(x): x is smart, D(x): x is destructive, C(x): x is cute, M(x): x is a tomcat, F(x): x is a female, T(x,y): x wants to mate with y ∃ xS(x): there exists a cat that is smart ∀ xC(x): All cats are cute ∃ [C(x)ΛD(x)]: There exists a cat that is cute and destructive x ∀ xS(x) -> ~D(x)]: All cats that are smart are not destructive ∀ x yM(x)ΛF(y) -> T(x,y)]: for all cats x, for all cats y, if x is a male tomcat and y is a female in heat, then x wants to mate with y U = U = U = ZΛ + (Positive Integers) x y z E(x): x is even, P(x): x is prime ∀ xE(x)Λ(x >= 4) -> ∃ ∃ [Py zΛP(z)Λ(X=y+z)]:for all evens >= 4, there are two prime numbers whose sum is equal to that even number

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