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# Discrete math indroductory week and beginning of lesson one CS 2305

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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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