Syllabus Every Week: 2 Hourly Exams +Final - as noted on Syllabus
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1 Syllabus Every Week: 2 Hourly Exams +Final - as noted on Syllabus Lecture Section: TuTh 9:30-10:45Lab/Discussion Section M & W 9-9:50 or 10-10:50Every Week:WorksheetQuizHomework2 Hourly Exams +Final - as noted on SyllabusWeb Page
2 Motivation Why Learn This Material?? Some things can be "directly applied"Some things are "good to know"Some things just teach "a way of thinking“ and expressing yourselfOverall Theme:Proofs
3 Course Content Propositional Logic (and circuits) Predicate Calculus - quantificationNumber TheoryMathematical InductionCounting - Combinations and ProbabilityFunctionsRelationsGraph Theory
4 Statement / Proposition declarativemakes a statementcan be understood to be either true or false in an interpretationsymbolized by a letterExamples:Today is Wednesday.5 + 2 = 73 * 6 > 18The sky is blue.Why is the sky blue?Brett FavreThis sentence is false.
5 Other Symbols and Definitions to make compound statements Conjunctionand --- symbolized by Disjunctionor symbolized by Negationnot symbolized by ~Truth Tables for these operatorsAloneCombinedExamples of Combined Truth Tables(p ^ r) (q ^ r)(p ^ ~r) (p ^ r)(p ^ q) (~q ~p)
6 Translation of English to Symbolic Logic Statements The sky is blue.one simple (atomic) statement - assign to a letter i.e. bThe sky is blue and the grass is green.one statementconjunction of two atomic statementseach single statement gets a letter i.e. b gand join with ^ i.e. b ^ gThe sky is blue or the sky is purple.disjunction of two atomic statementseach single statement gets a letter i.e. b pand join with i.e. b p
7 Trickier Translation 1 The sky is blue or purple. two statements the sky is blue assign this to bthe sky is purple assign this to pstill a disjunctionthe sky is blue or the sky is purpleb v p
8 Trickier Translation2 The sky is blue but not dark. two statements the sky is blue assign this to bthe sky is dark assign this to dconjunction with negationthe sky is blue and the sky is not darkthe sky is blue and it is not the case that the sky is dark"it is not the case that the sky is dark" is ~db ^ ~ d
9 Trickier Translation 3 2 x 6 English: x is greater than or equal to 2 and less than or equal to 6two statements:x is greater than or equal to assign this to px is less than or equal to assign this to qbecomesp ^ q
10 #3 Continued --- 2 x 6 p is actually a compound statement x is greater than 2 or x is equal to r sx is greater than 2 is symbolized by rx is equal to 2 is symbolized by sq is actually a compound statementx is less than 6 or x is equal to m nx is less than 6 is symbolized by mx is equal to 6 is symbolized by np ^ q becomes (r s) ^ (m n)
11 More about Operators exclusive or: p, q p or q but not both same as (p q) ^ ~(p ^ q)Precedence between the operators~ (not) highest precedence^ (and) / (or) have equal precedenceuse parentheses to override default precedencea ^ b cAMBIGUOUS(Alice and Bob went to the store) or Cal wentAlice went to the store and (either Bob or Cal went)
12 Special Results in the Truth Table Tautological Propositiona tautology is a statement that can never be falsewhen all of the lines of the truth table have the result "true"Contradictory Propositiona contradiction is a statement that can never be truewhen all of the lines of the truth table have the result "false"Logical Equivalence of two propositionstwo statements are logically equivalent if they will be true in exactly the same cases and false in exactly the same caseswhen all of the lines of one column of the truth table have all of the same truth values as the corresponding lines from another column of the truth tableExamples of Special Truth Tables (may be special)p ~pp ^ ~p(p ^ r) ^ ~(p r)(p ^ ~q) ? (~q ^ p)(p ^ q) ? ~ (~q ~p)
13 Logical Equivalences Theorem 1.1.1 - Page 14 Double Negative:~(~p) pCommutative:p q q p and p ^ q q ^ pAssociative:(pq)r p(qr) and (p^q)^r p^(q^r)Distributive:p^(qr) (p^q)(p^r) and p(q^r) (pq)^(p r)
14 More Logical Equivalences Idempotent:p ^ p p and p p pAbsorption:p (p ^ q) p and p ^ (pq) pIdentity:p ^ t p and p c pNegation:p ~p t and p ^ ~p cUniversal Bound:p ^ c c and p t tNegations of t and c:~t c and ~c t
15 DeMorgan's Laws ~( p q ) ~p ^ ~q ~( p ^ q ) ~p ~q It is not the case that Pete or Quincy went to the store. Pete did not go to the store and Quincy did not go to the store.It is not the case that both Pete and Quincy went to the store. Pete did not go to the store or Quincy did not go to the store.
16 Prove by Truth Table & by Rules ~(p ~q) (~q ^ ~p) ~p~((~p^q) (~p^~q)) (p^q) p(p q) ^ ~(p ^ q) (p ^ ~q) (q ^ ~p)~(p v ~q) v (~q ^ ~p) =(~p ^ ~~q) v (~q ^ ~p) by DeMorgan’s =(~p ^ q) v (~q ^ ~p) by DN =(~p ^ q) v (~p ^ ~q) by Commutative =~p ^ (q v ~q) by Distributive =~p ^ t by Negation Law =~p by Identity Lawfind typo in (~a ((a ^ b)^e)) ((a^~b)^~e) ~a ((b ^ e) ~(b e))
17 Conditional Statements Hypothesis ConclusionIf this, then that. Hypothesis implies Conclusion has lowest precedence (~ / ^ / )If it is raining, I will carry my umbrella.If you don’t eat your dinner, you will not get desert.pqp q1
18 Converting: to p q ~p q ~(p q ) p ^ ~q Show with Truth Table~(p q ) p ^ ~qShow with Truth Table and Rules
19 Contrapositive Negate the conclusion and negate the hypothesis Use the negated Conclusion as the new Hypothesis and the negated Hypothesis as the Conclusionp q ~q ~pEnglish:If Paula is here, then Quincy is here.If Quincy is not here, then Paula is not here.
20 Converse and Inverse p q q p ~p ~q Converse: Inverse: If Paula is here, then Quincy is here.Converse:swap the hypothesis and the conclusionq pIf Quincy is here, then Paula is here.Inverse:negate the hypothesis and negate the conclusion~p ~qIf Paula is not here, then Quincy is not here.
21 biconditionalp if and only if qp qp iff qpqp q1
22 Only If Translation in English Translation to if-then form p only if q p can be true only if q is trueif q is not true then p can't be trueif not q then not p (~q ~p)if p then q (p q)Translation in EnglishYou will graduate is CS only if you pass this course.G only if PIf you do not pass this course then you will not graduate in CS.~P ~GIf you graduate in CS then you passed this course.G P
23 Other English Words for Implication Sufficient Condition"if r, then s" r smeans r is a sufficient condition for sthe truth of r is sufficient to ensure the truth of sNecessary Condition"if y, then x" y xequivalent to "if not x, then not y" ~x ~ymeans x is a necessary condition for ythe truth of x is necessary if y is trueSufficient and Necessary Conditionp if, and only if q p qthe truth of p is enough to ensure the truth of q and vice versa
24 Argument A Sequence of Statements where The last in the sequence is the ConclusionAll others are Premises (Assumptions, Hypotheses)(premise1 ^ premise2 ^ …premiseN) conclusionCritical Rows of the Truth Tablewhere all of the premises are trueOnly one premise being false makes the conjunction falseA false hypothesis on a conditional can never make the whole falseThe Truth Value of the Conclusion in the Critical RowsValid Argument If and only if all Critical rows have true conclusionInvalid Argument If any single Critical row has a false conclusionp v q truth table has 3 critical rows with true conclusion and 1 with falseq v rp -> q(p^q) -> r
25 Rules of Inference (Table 1.3.1- Page 39) Modus Ponensp qpqModus Tollens~q~pDisjunctive Syllogismp q p q~q ~p p qDisjunctive Additionp q p q p qConjunctive Simplificationp ^ q p ^ qRule of Contradiction~p cpHypothetical Syllogismq rp rConjunctive Additionq p ^ qDilemmap qp rrPractice: 1) p ^ q | p r >> r ^ q valid2) p | q p >> q invalid3) p v q |q r | ~p --->> r valid
26 Proofs Using Rules of Inference p qP2q rP3~prP1p ^ qP2p sP3~r ~qs ^ rP1p qP2~(q r)P3p (m r)~m
27 Conditional WorldsMaking Assumptions - Only Allowed if you go into a "conditional world"list of statements that are true in all worlds|| Assume anything| list of statements true| in the worlds where the assumption is true|Assumption -> anything from the conditional world
28 Conditional World Assumption Leads to Contradiction Make an Assumption, but that Assumption leads to a contradiction in the conditional world.list of statements that are true in all worlds|| Assume anything| list of statements true| in the worlds where the assumption is true| a contradiction with something else known true|Assumption must be false in all possible worlds