1 Propositional Logic lDeS A pumposition is a sentence that declares a fact that is either true or false but not both Eg Which of the following are propositions Berkeley is a city in the US true prop 3 It I false prop Read this carefully X not declarative Xt y Z declares somethings that's neither true X false nor R P Q Use letters to denote pnposi variables variables i.e that represent propositions or false CF of a proposition is true CT The truth based on whether the proposition is a true proposition or not We can form new propositions from the old ones Q IDefI Let p of be proplositions 7 P C or P The negation of p denoted is the prop a 11 is not the case that P It The conjunction of p and Q denoted p AQ is the prop u p and d n
The disjunction of P and Q denoted P V Q is the prop P or Q EI we all love discrete math n we all love probability theory T a true prop since both clauses are true N V you can think of them as functions for Tuthtat 7 n p p v Q a p na p T T T F T T F F F T F T T T F F F F F T IDefI Let p d be prop hypothesis conclusion Q is the prop 4 if P then Q imply The p The biconditional p Q is the prop up if and only if Q P P Q a Q w p g T T 11 1 2 is necessary for Yiningtolike math T Ctf 3 F Yining Yikes math on T F a UPnicorns is sufficient for ft F existence of T The T 2 Q W PW xI fe You can g fmisa Fdeducetanythin
Propositional Equivalence 1.2 IDefI A compound prop that is always true regardless of the truth values of the prop Variables that occur in it is called a last column in the truth table only has 1 tautology ie i p v EI p p T T F T so P v n p is a tautology IDefI Two compound prop p and Q are logicaliyequivalent PE Q is a tautology denoted p if Q i.e the last columns match EI Prd n p v n Q Q Q p n p F F T T T T F T F T T T T T F T F T F F T F T F distribute and flip DeMorganislai n p v n Q Pnd n p n n Q Pva
GP P Q Eg d n p v Q n p Q P Q p T F T T T F F T F F F T T T T T T F F T E Which one is correct Rem converse Q a p contrapositive Go sp a p n p n p nd Q nd p Q p Q p T T T F F T T T F F T F F T F T F T T T F F F T T T T T c E.ge I'm Yining I don't own an unicorn I'm Yining I don't own an unicorn converse I'm not Yining an unicorn I contrapositive own
2 First Order Logic Recoil Let Plx be the statement X 3 because PGD doesn't have a P X a prop is not truth value PLK is called a prepositional function Once we assign value to P X becomes a proposition X E.ge Let P X y be the propositional function X y Then PC o D has truth value F PC 3 2 has truth value T Another way to create a proposition from propositionalfunction involves quantifiers IDefI The domain of a propositional function p is x the set of all possible values of oc IDefI We use to denote for eachvalue of PCX pix in the domain X to denote there exists one element we use in the domain such that P Lx x
Reem We often indicate the domains of x using KES theset of natural numbers 0,1 2,3 what's the truth value of EI f x c IN T 7 M X 30 KE IN I T X 30 distribute DeMorgan's Laws flip Fx pix FX apex 7 F x P FX x np the same as Fy tx Peay Reem Is Footy Pac y No I a co I f stronger a X Y X Y Assume we're in our universe with more than one person and EI more than one unicorn Voce unicorns Fy c human y owns x each unicorn has i e owner a If ye human txt unicorns y owns x there's someone who owns all unicorns i e
Fun One of us built Join Me Amin Not me 7A Not me Khalil 7 K Yining Amin built it A Only one of us is telling the truth Who built Join Me Y A K Amin built it n A aka Ann H we know Kha Y v Anak n Y v n KMA An kn A AA and V A AKA A v TANK Ark v iF TO
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