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Discrete math indroductory week and beginning of lesson one

by: Aaron Maynard

Discrete math indroductory week and beginning of lesson one CS 2305

Marketplace > ComputerScienence > CS 2305 > Discrete math indroductory week and beginning of lesson one
Aaron Maynard
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These notes cover the start of the section over proposition. In the introductory (14th) week, we learned about the symbols and meaning of common propositions. In week of January 21st, we started co...
Discrete Math for Computing I
Timothy Farage
Class Notes
Math, Discrete math, Computer Science, Computer Science I, University of Texas at Dallas
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This 4 page Class Notes was uploaded by Aaron Maynard on Saturday January 16, 2016. The Class Notes belongs to CS 2305 at a university taught by Timothy Farage in Spring 2016. Since its upload, it has received 52 views.

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Date Created: 01/16/16
Discreet Math for Computing Aaron Maynard Timothy Farage January 14 , 2016 There are two parts to math. MATH LOGIC SET THREORY Fermat's Last Theorem: + n n n Let A, B, C, N € Z exist such that if A + B = C , then n ≤ 2 LOGIC Definition: A proposition is a statement "This is either True or False". Examples 2 + 3 = 7 True All cats have hearts True X + 2 = 7 False “There is an ‘x’ such that ‘x + 2 = 7” True Logic Variables are either True (1) or False (0) It is traditional to use variables such as P, Q, R1 o2 Pn, P , P . Logical Operators (Boolean Operators) Also known as an unary operator, only deals with one proposition. P !P (not P) 0 1 1 0 Binary Operators  PVQ, pronounced "P or Q" or "P or Q or Both”  P ΛQ, pronounced "P and Q"  P⊕Q / {P xor Q, pronounced "P x or Q" or "P or Q but NOT both"  P<->Q, pronounced "P if and only if Q"  P->Q, pronounced "If P then Q" or "P implies Q" Discreet Math for Computing Aaron Mthnard Timothy Farage January 14 , 2016 Truth Table for the Binary Operators P Q PVQ P ΛQ P⊕Q P<->Q P->Q 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 Side Notes from Class  Example of P<->Q: o Two sides of a triangle are equal if and only if the angles opposite those sides are equal. o If two sides of a triangle are not equal then the angles opposite those sides are not equal.  P is called the antecedent, Q is called the consequence / conclusion. Discreet Math for Computing AarothMastard Timothy Farage January 19 -21 , 2016 Compound Proposition Definition: Using multiple propositions with apparition. P -> (Q V R) P Q R ( Q V R ) P -> ( Q V R ) 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 If every circumstance is true, then the compound proposition is known as an example of "Tautology" A logical expression that is true for any values of its variables is said to be tautology. ( P Λ Q ) Λ (P -> Q) P Q PΛQ P->Q (PΛQ) Λ (P- >Q) 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 A compound proposition that is false for all values of its variables is known as a "contradiction". “I am that I am." Discreet Math for Computing AarothMastard Timothy Farage January 19 -21 , 2016 Math Equivalences: For any x and y, x + y = y + x x + 2 = 5 Logical Equivalences (Propositional Equivalences):  PVQ  QVP  PΛQ  QΛP  (PVQ)VR  PV(QVR)  PΛ(QVR)  (PΛQ)V(PΛR) ~P is known as “not P” Negate and Simplify: Definition: Simplifying until there are no negations except over variables. ~[(PΛQ) -> (Q->R)]  (PΛQ) Λ ~(Q->R)  (PΛQ) Λ (QΛ~R)  PΛQΛ~R Done! Converse of P -> Q If the animal is a cat, then it has four legs. Q -> P If the animal has four legs, then it is a cat. If a triangle has 2 equal sides, then it has 2 equal angles. If a triangle has 2 equal angles, then it has 2 equal sides. Contrapositive of P -> Q If you get over 90%, then you get an A. ~P -> ~Q If you get an A, then you get over 90%. Things to have memorized: P->Q <=> ~PVQ ~(P->Q) <=> P Λ ~Q - The negation of an implication is NOT an implication, it's an AND! Demorgans Laws: ~(PVQ) <=> ~P Λ ~Q ~(P Λ Q) <=> ~PV~Q


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