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# DISCRETEMATHEMATICS MATH245

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Math 245 Discrete Mathematics The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments Lecture Notes 3 Peter Blomgren Department of Mathematics and Statistics San Diego State University San Diego CA 921827720 blomgrenterminusSDSUEDU httpterminusSDSUEDU Id lectureitexv 116 20060907 182434 blomgren Exp The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 160 Previouslyz Logical Form and Equivalence 1 of 2 Statements Sentences that are either TRUE or FALSE but not both Logical symbols N not and or Statement form An expression made up of statement variables and symbols that becomes a statement when actual statements are substituted for the state ment variables The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 260 Previouslyz Logical Form and Equivalence 2 of 2 Truth table A table showing all possible truthvalue combinations of the statement variables p q r as well as the corresponding truth values for a simple or compound statement of interest In the case of a compound statement we also tend to include columns for intermediate statements Logical equivalence Two logical expressions with the same truth values columns in a truth table are said to be logically equivalent ie two different ways of expressing the same thing Tautology A logical expression that is always true for all input logical variables Eg p N p Contradiction A logical expression that is always false for all input logical variables Eg p N p The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 360 Conditional Statements if then gt A logical inference or deduction is made from a hypothesis to a conclusion Let p and q be statements A sentence of the form if p then q is denoted by 19 p is the hypothesis and q the conclusion gt is a logical connective and like A N and it can be used to join statements to create new statements To define p gt q as a statement we must specify the truth values for p gt q just as we did for p q and friends The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 460 If Then gt Truth Table The formal definition of truth values for gt is based on its everyday intuitive meaning The promise If you show up for class on Tuesday then you will get an A in this class is false only if you do show up for class on Tuesday and do not get an A in this class In all other cases it is true the promise is not broken Hence the truth table looks like n n l lhs C T F T F l lquotn ll The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 560 Example Truth Table for p V N g gt N p Recall Definition Conditional lfp and q are statement variables the conditional of q by p is if p then q or p implies q and is denoted p gt q It is false when p is true and q is false otherwise it is true Np 61 pVQ pV 1 p n n l lhe l39l l l39l vQ The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 660 Example Truth Table for p V N g gt N p Recall Definition Conditional lfp and q are statement variables the conditional of q by p is if p then q or p implies q and is denoted p gt q It is false when p is true and q is false otherwise it is true Np 61 pVQ pV 1 p n n l lhe l39l l l39l vQ l l r39l n The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 660 Example Truth Table for p V N g gt N p Recall Definition Conditional lfp and q are statement variables the conditional of q by p is if p then q or p implies q and is denoted p gt q It is false when p is true and q is false otherwise it is true Np 61 pVQ pV 1 p n n l lhe l39l l l39l vQ l l r39l n l r39I ln The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 660 Example Truth Table for p V N g gt N p Recall Definition Conditional lfp and q are statement variables the conditional of q by p is if p then q or p implies q and is denoted p gt q It is false when p is true and q is false otherwise it is true Np 61 pVQ pV 1 p n n l lhe l39l l l39l vQ l l r39l n l r39I ln The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 660 Example Truth Table for p V N g gt N p Recall Definition Conditional lfp and q are statement variables the conditional of q by p is if p then q or p implies q and is denoted p gt q It is false when p is true and q is false otherwise it is true p q Np 61 pVQ PVNQ gtNP T T F F T F T F F T T F F T T F F T F F T T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 660 Vacuously True True By Default A conditional statement p gt q that is true by virtue of the fact that the hypothesis p is false is often called vacuously true or true by default The statement If you show up for work on Tuesday morning then you will get thejobquot is vacuously true if you do not show up for work on Tuesday morning In this case there is no promise hence it cannot be broken The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 760 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 qu p W anquot quHT p gtTq gtT 39n39n39n39n i i i i 39n39nll39n39n l I Q 39n l39n l39n I39n Is The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 p q 7quot PVQ p W q W INCH gt7 pewq W T T T T T T F T T F T T T F F T F T T T F T F T F F T F F F F F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 1 pvqnr pewqer Q V 8 19 T1 T1 T1 T1 1 1 1 1 8 T1 T1 l l T1 T1 l Q 3911 1 39n 1 39n 1 39n 1 a 3911 39n 1 1 1 1 1 1 lt 1 1 1 1 39n 1 39n 1 i The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 p q 7quot PVQ p W q W INCH gt7 pewq W T T T T T T T T F T F F T F T T T T T F F T F T F T T T T T F T F T T F F F T F T T F F F F T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 p q 7quot PVQ p W q W INCH gt7 pewq W T T T T T T T T T F T F F F T F T T T T T T F F T F T F F T T T T T T F T F T T F F F F T F T T T F F F F T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Logical Equivalences Involving gt Example Showing that p q gt 7 E p gt 7 q gt 7 p q 7quot PVQ p W q W INCH gt7 pewq W T T T T T T T T T T F T F F F F T F T T T T T T T F F T F T F F F T T T T T T T F T F T T F F F F F T F T T T T F F F F T T T T Since the last two columns match we have shown that pV q gtTEp gtT CI gt7 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 860 Negation of a Conditional Statement The negation of if p then q is logically equivalent to p and not q Proof p gtq 61 NOD gt61 pq n n l lhe l39l l l39l vQ The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 960 Negation of a Conditional Statement The negation of if p then q is logically equivalent to p and not q Proof 29 q p gtq 61 NOD gt61 pANCJ T T T T F F F T T F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 960 Negation of a Conditional Statement The negation of if p then q is logically equivalent to p and not q Proof 29 q p gtq 61 NOD gt61 pANCJ T T T F T F F T F T T F F F T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 960 Negation of a Conditional Statement The negation of if p then q is logically equivalent to p and not q Proof 29 q p gtq 61 NOD gt61 pANCJ T T T F F T F F T T F T T F F F F T T F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 960 Negation of a Conditional Statement The negation of if p then q is logically equivalent to p and not q Proof 29 q p gtq 61 NOD gt61 pANCJ T T T F F F T F F T T T F T T F F F F F T T F F Example Note that we use N N q E g N If my car is in the shop then I cannot get to class E My car is in the repair shop and I can get to class The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 960 The Contrapositive of a Conditional Statement Definition Contrapositive The contrapositive of a conditional statement of the form if p then q is If lt a then lt pquot Symbolically the contrapositive of p gt g is q gt N You will be asked see homework to show that A conditional state ment is logically equivalent to its contrapositive ie iv gt61E 61 a N 29 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1060 Examples Writing the Contrapositive 1 The contrapositive of If Howard can swim across the lake then Howard can swim to the island quot is If Howard cannot swim to the island then Howard cannot swim across the lake 2 The contrapositive of If today is Easter then tomorrow is Monday quot is If tomorrow is not Monday then today is not Easterquot The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1160 The Contrapositive is an Important Tool We will see the contrapositive form later on in this class The logical equivalence of a conditional statement and its contrapos itive is the basis for one of the laws of deduction modus tollens and for the contrapositive method of proof The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1260 The Inverse and Converse of a Conditional Statement Definition Converse and Inverse Suppose a conditional statement of the form if p then q is given 1 The converse is if q then p The inVerse is if N then N q Symbolically The converse of p a q iS q H P The inverse of 19 q iS N P m N 9 Note The inverse and converse are not logically equivalent to the statement they are however logically equivalent to each other since the inverse is the contrapositive of the converse Midterm alert The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments p 1360 Only if To say p only if q means that p can take place only if q takes place also That is if q does not take place then p cannot take place By the logical equivalence of the contrapositive we can also say that ifp occurs then q must also occur Definition Only If If p and q are statements p only if q means if not q then not p or equivalently if p then qquot The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1460 If and only ifquot The Biconditional Definition If and Only If Given the statement variables p and q the bi conditional of p and q is p if and only if q and is denoted p lt gt Q It is true if both p and q have the same truth values and is false if p and q have opposite truth values The words if and only if are sometimes abbreviated iff P C PHCJ T T T T F F F T F F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1560 Order of Operations In order of operations lt gt is co equal with gt and we have the following precedence for our five logical connectives highest 1 N l 2 v lowest 3 gt lt gt Order of operations The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1660 if only ifquot and if and only ifquot According to the definitions of if and only if saying p if and only if q should mean the same as saying p if q and p only if q That is indeed the case again we look at the truth table ponlyifq pifq piqu ponlyifqandpifq iv gt61 CI gt29 pea p gtCJMq gtp n n l lte n l n Q The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1760 if only ifquot and if and only ifquot According to the definitions of if and only if saying p if and only if q should mean the same as saying p if q and p only if q That is indeed the case again we look at the truth table p only if q p if q p ifF q p only if q and p if q p q iv gt61 Q gtP pea iv gt61Aq gtp T T T T F F F T T F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1760 if only ifquot and if and only ifquot According to the definitions of if and only if saying p if and only if q should mean the same as saying p if q and p only if q That is indeed the case again we look at the truth table p only if q p if q p ifF q p only if q and p if q p q iv gt61 Q gtP pea iv gt61Aq gtp T T T T T F F T F T T F F F T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1760 if only ifquot and if and only ifquot According to the definitions of if and only if saying p if and only if q should mean the same as saying p if q and p only if q That is indeed the case again we look at the truth table p only if q p if q p ifF q p only if q and p if q p q iv gt61 Q gtP pea iv gt61Aq gtp T T T T T T F F T F F T T F F F F T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1760 if only ifquot and if and only ifquot According to the definitions of if and only if saying p if and only if q should mean the same as saying p if q and p only if q That is indeed the case again we look at the truth table p only if q p if q p ifF q p only if q and p if q p q iv gt61 Q gtP pea iv gt61Aq gtp T T T T T T T F F T F F F T T F F F F F T T T T Since the last two columns are equal the statement forms are equiv alent ie p lt gt q E p gt q q gt p The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1760 Necessary and Sufficient Conditions The phrases necessary condition and sufficient condition as used in formal English correspond exactly to their definitions in logic Definition Sufficient and Necessary Conditions If r and s are statements r is a sufficient condition for 3 means if r then s r is a necessary condition for 3 means if not r then not 3 Note that due to the equivalence between a statement and its contra positive r is a necessary condition for s also means if s then r The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1860 Solved Problems Epp1228 1 of 2 Eppl 2 28 Do you mean that you think you can find out the answer to it said the March Hare Exactly so said Alice Then you should say what you mean the March Hare went on I do Alice hastily replied at least at least I mean what I say that39s the same thing you know Not the same thing a bit said the Hatter Why you might just as well say that I see what I eat is the same thing as I eat what I see from A Mad Tea Party in Alice in Wonderland by Lewis Carroll That Hatter is right I say what I mean is not the same thing as I mean what I say Rewrite in if then form and explain the difference The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 1960 Solved Problems Epp1228 2 of 2 The if then form of I say what I meanquot is quotIf I mean something then I say it quot mean gt say The if then form of I mean what I say is quotIf I say something then I mean it quot say gt mean The two statements are the converse of each other and are not logically equivalent Corresponds to Eppv201224 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2060 Homework 1 Due 9152006 1200pm GMCS587 Epp l2 13 24 25 26 27 Eppll 3 l4 16 21 25 29 31 37 41 Extra BrainTwister for fun Epp1154 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2160 Arguments Introduction We are now going to use our new tools language logic statements connectives conditionals to generate arguments In mathematics logic an argument is not a dispute rather Definition Argument An argument is a sequence of statements All statements but the final one are called premises or assumptions or hypotheses The final statement is called the conclusion The symbol read therefore is normally placed just before the conclusion We will be concerned with determining whether an argument is valid that is to determine whether the conclusion follows necessarily from the preceding statements The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2260 Abstracting the Content from the Arguments We have already seen Lecture Notes 2 that we can separate the content from the argument recall Statement A If Jane is a math major or Jane is a CS major then Jane will take Math 245 Jane is a CS major Therefore Jane will take Math 245 Abstract logical form With our new symbol If p or q then r If p or q then r q q Therefore r r The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2360 Valid Arguments When we consider the abstract form of an argument eg If p or q then r q r we think of p q and r as variables for which statements may be substituted Definition Valid Argument Form To say that an argument form is valid means that no matter what particular statements are substituted for the statement vari ables in its premises if the resulting premises are all true then the conclusion is also true To say that an argument is valid means that its form is valid The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2460 Valid Arguments The truth of the conclusion of a valid argument follows necessarily or inescapably or by logic alone from the truth of its premises It is impossible to have a valid argument with true premises and a false conclusion When an argument is valid and its premises are true the truth of the conclusion is said to be inferred or deduced from the truth of the premises The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2560 Testing for Validity To test an argument form for validity 1 Identify the premises and conclusion of the argument 2 Construct a truth table showing the truth values of all the premises and the conclusion 3 Find the critical rows in which all the premises are true 4 In each critical row determine whether the conclusion of the argument is also true a If in each critical row the conclusion is also true then the argument form is valid b If there is at least one critical row in which the conclu sion is false the argument form is invalid The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2660 Example Time Show that the following argument form is valid 29 V q V 7 N 7 PVQ variables premises conclusion qVT pVqVT N qu n n n n l l l l n n l l n n l IQ n l n l n l n l The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time Show that the following argument form is valid 29 V q V 7 N7 PVQ variables premises conclusion pVqVT N qu V n n n n l l l l n n l l n n l IQ n l n l n l n l n l l l n l l llt The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time Show that the following argument form is valid 29 V q V 7 N7 PVQ variables premises conclusion pVqVT N qu V n n n n l l l l n n l l n n l IQ n l n l n l n l n l l l n l l llt n l l l l l l l The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time Show that the following argument form is valid 29 V q V 7 N 7 29 V 61 variables premises conclusion 29 q 7 qVT pVqV I N qu T T T T T F T T F T T T T F T T T F T F F F T T F T T T T F F T F T T T F F T T T F F F F F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time Show that the following argument form is valid 29 V q V 7 N 7 29 V 61 variables premises conclusion 29 q 7 qVT pVqV I N qu T T T T T F T T T F T T T T T F T T T F T T F F F T T T F T T T T F T F T F T T T T F F T T T F F F F F F F T F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time A Valid Argument Form Show that the following argument form is valid 29 V q V 7 N 7 29 V 61 variables premises conclusion 29 q 7 qVF pVqVF N qu T T T T T F T T F T T T T T F T T T F T F F F T T T F T T T T F F T F T T T T F F T T T F F F F F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2760 Example Time Show that the following argument form is invalid p gtqr q gtp7 P W variables premises conclusion p q 7 N7 qVNT pAr p qvwr q gtpr p gtr T T T T T F T F T T F F F T T F T F F F T F F F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time Show that the following argument form is invalid p gtqr q gtp7 P W variables premises conclusion p q 7 N7 qVNT pAr p qvwr q gtpr p gtr T T T F T T F T T F T F T F F T F T T F F T F T F F T F F F F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time Show that the following argument form is invalid p gt C N 7 q gt p 7 P W variables premises conclusion 2 7 qVNT pAr p qvwr q gtpr DQ p W 39n39n39n39n l l l l B 39n39n l l39n39n l l 39n I39n I39n l39n l s l39n l39n l39n ln l39n l l l39n l l The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time Show that the following argument form is invalid p gtq N 7 q gtp 7 P W variables premises conclusion p q 7 N7 qVNT pAr p qvwr q gtpr p gtr T T T F T T T T F T T F T F T F F T T F F T T F F T T F T F F T F T T F F F T F F F F F F T T F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time Show that the following argument form is invalid p gtq N 7 CI gtP 7 29 variables premises conclusion p q 7 N7 qVNT pAr p qvwr q gtpr p gtr T T T F T T T T T F T T F T T F T F F T F T F F T T F T F T T F T F T F T F T T F T F F T F F F T F F F T T F T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time Show that the following argument form is invalid P gtCJV N 7 CJ gtP 7 P W variables premises conclusion p q 7 N7 qVNT pAr p qvwr q gtpr p gtr T T T F T T T T T T F T T F T F T F T F F T F T T F F T T F T T F T T F T F T F F T F T T F T F F F T F F F T T F F F T T F T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Example Time An Invalid Argument Form Show that the following argument form is invalid p gtq N 7 q gtp 7 P W variables premises conclusion p q r Nr qvwr pr p qvwr q gtpr p gtr T T T F T T T T T T T F T T F T F T F T F F T F T T F F T T F T T F F T T F T F T F F T F T T F T F F F T F F F T T T F F F T T F T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2860 Modus Ponens The Method of Affirming If we have an argument of the form If p then q p q The fact that this argument forms is valid is called modus ponens from Latin premises conclusion p C p gt CI p C T T T T T T F F T F T T F F F T F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 2960 Modus Tollens The Method of Denying If we have an argument of the form If p then q If Nq then N p N q contragositive N q 39 N p N p The fact that this argument forms is valid is called modus tollens from Latin premises conclusion p C p gt C N C N p T T T F T F F T F T T F F F T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3060 Disjunctive Addition Generalization Disjunctive addition is used for making generalizations P CI 10 V 61 10 V 61 premises conclusion premises conclusion I9 q p I9 V q p q q I9 V q T T T T T T T T T F T T T F F F T F F T T T F F F F F F Example Students p and LOGICAL 0R Seniors q get a discount at store X You are a student therefore p q you get a discount The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3160 Conjunctive Simplification Specialization Conjunctive simplification is used for particularizing P Q P Q P Q premises conclusion premises conclusion P Q P A Q P P Q P A Q Q T T T T T T T T F F T F F F T F F T F F F F F F F Example You are tired of logic and Peter Therefore in particular you are tired of logic The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3260 Disjunctive Syllogism Elimination Disjunctive Syllogisms are used to rule out possibilities 10 V 61 10 V 61 N C N P P C1 premises conclusion premises conclusion 29 q 29 V q q p p q 29 V q N p q T T T F T T T F T F T T T T F T F F T T F F T T T T F F F T F F F T Example You are tired of logic or surfing You are not tired of surfing Therefore you are tired of logic The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3360 Hypothetical Syllogism Transitivity Hypothetical Syllogisms are used to build chains of implication p gt C q gt 7 p gt 7quot premises conclusion 29 q 7 p gt q q gt 7 p gt 7 T T T T T T T T F T F T F T F T F F F F T T T T T F T F T F F F T T T T F F F T T T The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3460 Hypothetical Syllogism Example If it is sunny the sky is blue Ifthe sky is blue we ll go surfing Therefore If it is sunny we ll go surfing sunny gt sky blue sky blue gt surfing sunny gt surfing The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3560 Where are the glasses A Complex Deduction 1 of 2 The following statements are true a If my glasses are on the kitchen table V p then I saw my glasses at breakfast p gt q V q b I was reading the newspaper in the living room V 7 or I was reading the newspaper in the kitchen r V s 8 c If r then my glasses are on the coffee table r gt t t d I did not see my glasses at breakfast N q e If I was reading my book in bed V u then my glasses are on the bed table u gt v V U f If s then p s gtp The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3660 Where are the glasses A Complex Deduction 2 of 2 We have the following ap gtq b7 8 r Hf c d Nq eu gtv fs gtp We make the following deductions 1 By a and d we deduce N p by modus tollens 2 By E and 1 we deduce N s by modus tollens 3 By b and 2 we deduce by disjunctive syllogism 4 By c and 3 we deduce by modus ponens Hence the glasses are on the coffee table The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3760 Fallacies Broken Logic A fallacy is an error in reasoning resulting in an invalid statement Three common mistakes 1 Using vague or ambiguous premises 2 Assuming what is to be proved 3 Jumping to conclusions without adequate grounds In the next few slides we39ll explore two other fallacies 4 Converse Error 5 lnverse Error Which give rise to arguments which resemble modus ponens and modus tollens but are invalid The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3860 Checking for Fallacies There are two ways 1 Construct the truth table and demonstrate that there is at least one critical row in which the premises are true but the conclusion false 2 Find an argument of the same form logical equivalence with true premises and a false conclusion Counterexample The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 3960 Converse Error If Peter is a cheater then Peter will sit in the back row Peter sits in the back row Therefore Peter is a cheater It is quite possible that Peter is not a cheater but is sitting in the back row You will be asked homework to construct the truth table showing that this type of argument is invalid The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4060 Inverse Error If Peter is a cheater then Peter will sit in the back row Peter is not a cheater Therefore Peter does not sit in the back row It is quite possible that Peter is not a cheater even though he is sitting in the back row You will be asked homework to construct the truth table showing that this type of argument is invalid The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4160 Homework 1 Due 9152006 1200pm GMCS587 Epp13 13 21 39 Epp13 Read examples 1315 1316 Epp12 13 24 25 26 27 Epp11 3 14 16 21 25 29 31 37 41 Extra BrainTwister for fun Epp1154 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4260 Application of Logic Digital Circuits Introduction A lot of the theory of symbolic logic we have seen so far was developed by Augustus De Morgan 1806 1871 and George Boole 1815 1864 in the 19th century One of the cleanest application of logic in the wild is to construc tion of digital logic circuits In essence a processor chip is nothing but a huge collection of AND OR and NOTswitches Claude Shannon 1916 2001 made the connection between switched systems and logic and used formal logic to solve circuit design prob lems His master39s thesis A Symbolic Analysis of Relay and Switching Circuits was published in 1938 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4360 Application of Logic Digital Circuits Introduction Claude Shannon39s doctoral thesis was on theoretical genetics His paper A Mathematical Theory of Communication 1948 founded the subject of information theory The idea that one could transmit pictures words sounds etc by sending a stream of 139s and 039s down a wire was fundamentally new In 1956 William Bradford Shockley 1910 1989 John Bardeen 1908 1991 and Walter Houser Brattain 1902 1987 received the Nobel Prize in Physics for their researches on semiconductors and their discovery of the transistor effect quot The transistor is the small semiconductor device which makes modern computers possible The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4460 The Transistor We39ll take a quick look at how to build logic circuits using the transistor as a building block First let39s look at the transistor A bipolar junction transistor consists of three regions of doped semiconductors A small current in the center or base Collector Base Base region can be used to control a larger current flowing between F39NP NPN the end regions emitter and Cullectur Enlisctor collector The dewce can Ease Base be characterized as a current amplifier having many appli Emitter Emitter cations for amplification and switching quot Note Figures and text borrowed from httphyperphysicsphyastrgsuedu The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4560 The Transistor AND Gate A B 51 The use of transistors for the con ZHEZZZ tup struction of logic gates depends upon their utility as fast switches When the baseemitter diode is turned on enough to be driven into saturation the collector voltage with respect to ground may be less than a volt and can be used as a logic 0 in the TTL logic familyquot Here if we connect a true value or 6V to both A and B then both transistors open and the out value is Otherwise there is no connection to 6V from the out hence the value is false 0quot or 0V The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4660 The Transistor OR Gate A V B 51 The use of transistors for the con ZHZEZZ tun A their utility as fast switches When struction of logic gates depends upon quot5 the baseemitter diode is turned on 1m enough to be driven into saturation the collector voltage with respect to ground may be less than a volt and 4m can be used as a logic 0 in the TTL logic familyquot Here if we connect a true value or 6V to at least one of A and B then there is a path from out to 6V and the output value if true Otherwise there is no connection to 6V from the out hence the value is false or 0V The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4760 The Transistor NAND Gate N A B The use of transistors for the con struction of logic gates depends upon their utility as fast switches When EHZZEZ tup the baseemitter diode is turned on enough to be driven into saturation the collector voltage with respect to ground may be less than a volt and can be used as a logic 0 in the TTL logic family quot Here if we connect a true value or 6V to both A and B then both transistors open and the out value is Otherwise there is no connection to 0V from the out hence the value is true or 6V The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4860 The Transistor NOR Gate N A V B The use of transistors for the con 61quot struction of logic gates depends upon 1 4 32222 their utility as fast sWItches When I o quotL But the baseemitter diode is turned on 5 1K enough to be driven into saturation the collector voltage with respect to II B ground may be less than a volt and can be used as a logic 0 in the TTL logic family quot Here if we connect a true value or 6V to at least one of A and B then there is a path from to to 0V and the out value if false Otherwise there is no connection to 0V from the out hence the value is true or 6V The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 4960 Standard Circuit Symbols a QR D AND Q XOR A g NOT NOR To the left we see the standard circuit symbols for common logical connectives Note that we can build the missing ones XOR and NOT from the ones we already have OR AND NANDNOR The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 5060 Building 3 NOT circuit By connecting the input P to both ineports on the NANDegate we get an inverter NOTegate m m m 7psaso Building an XOR circuit PORQANDNOTPANDQ p Q 29 q qu pm pACJ PVQNPCJ T T T T F F T F T F T T F T T F T T F F F F T F The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 5260 Finding the Logic Boolean Expression for a Circuit PT PVQII AlP le In order the find the expression for a circuit for each gate simply apply the appropriate operation to the inputs The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 5360 The InputOutput Table for a Circuit The InputOutput table for a circuit is a table much like the truth table which shows the output value of the circuit for all possible combinations of inputs Two circuits are equivalent if and only if their inputoutput tables are identical 9 H39R Q Example of two equivalent circuits m m m trpsASO Showing that Two Circuits are Equivalent b Example of two equivalent circuits We can either construct the inputoutput tabies for the circuits and check that the tabies are identicai or we can use our knowiedge of symboiic iogic For the circuit above P N P Q E distributive law P N Q Q E negation law P t Q E P Q identity law The Logic of Compound Statements Conditional Statement Valid and Invalid Argument 7 p 5560 Adding Bits with Circuits P Q Carry Sum l 1 1 O l 0 0 1 0 1 0 1 0 0 0 0 The halfadder When adding binary bits we have the following in base 2 OOHH OHOH 7 p 5mm Adding More Bits with Circuits The FullAdder FULLADDER Circuit halfeaddcr 2 I S npuUOutput Table P Q R c s 1 l I l I 1 l 0 l 0 0 1 l 0 l 0 0 0 0 1 l 1 0 0 1 0 0 0 0 1 0 l 0 0 0 0 0 The Logic of Compound Stetemente ondxhonal Statement Valid and Invalid Axgnmente e p 5760 Homework 1 Due 9152006 1200pm GMCS587 Read sections 14 and 15 for background and entertainment value Epp1426 Epp1428 Suggested but not due Epp13 13 21 39 Epp13 Read examples 1315 1316 Epp12 13 24 25 26 27 Epp11 3 14 16 21 25 29 31 37 41 Extra BrainTwister for fun Epp1154 The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 5860 Dictionary From wwwwebstercom conjunction n a complex sentence in logic true if and only if each of its components is true disjunction n a compound sentence in logic formed by joining two simple statements by or syllogism n a deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion as in every virtue is laudable kindness is a virtue therefore kindness is laudable The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 5960 Homework 3rd Edition lt gt 2nd Edition 3rd Edition 2nd Edition Problems 11 3 14 16 21 25 29 31 37 41 11 3 12 14 19 23 27 29 35 37 12 13 24 25 26 27 12 13 20 21 22 23 13 13 21 39 13 12 20 38 14 26 28 14 26 28 Examples 138 138 1315 1316 1314 1315 Please use the 3rd Edition numbering when handing in your solutions The Logic of Compound Statements Conditional Statements Valid and Invalid Arguments 7 p 6060 C C Logic Minimization of Expressions Karnaughmaps and the Quine McCluskey Method Lecture Notes 3 Peter Blomgren Department of Mathematics and Statistics San Diego State University San Diego CA 921827720 blomgrenterminusSDSUEDU httpterminusSDSUEDU Id lectureitexv lill 20060919 222201 blomgren Exp Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 133 C One major application area of logic is circuit design For many reasons mostly related we want to make our circuits as efficient as possible Consider the following logically equivalent statements xAyAZVxAyAZ E yVyAxAZ txz xAz Clearly the expression x z is the most ef cient representation of this particular statement In this lecture we will explore two methods for such minimization of logic expressions Karnaugh maps Kmaps and the Quine McCluskey method Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 233 Kmaps is a graphical method for minimizing logic expressions by hand Kmaps are suited for expressions containing no more than six statement variables We consider an expression with two statement variables x and y There are four possible andnot minterms of x and y xAy xy xy xy A Kmap in these variables consists of four cells ywy where a 1 is placed in a cell if that minterm is part of the expression Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 333 The Kmaps for a 93 A y V N 93 A y b 56 A N y V 93 A y and C 56 A N y V 93 A y V 93 A N y are given by y N y y N y y N y x 1 x 1 x 1 N x 1 N x 1 N x 1 1 a b C Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 433 The Kmaps for a 93 A y V N 93 A y b 56 A N y V 33 A y and C 56 A N y V 33 A y V 33 A N y are given by 9 V y 9 quotV y 9 quotV y x 1 x 1 x 1 N x 1 N x 1 N x 1 1 a b C Whenever two adjacent cells in the Kmap have ls the minterms representing these cells can be combined into an expression involving just one of the variables Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 433 y quot y y X1 X xl1J x a b C The each block in the Kmaps above reduce to one expression so that the simplified expressions are a y b 93 A N y V N 93 A y C N 93 V N y Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 533 yz yz yz yz NX In three variables we set up a 4X2 rectangle of 8 cells Adjacent cells should differ in only one statement variable The leftmost and rightmost cells should be considered adjacent think of wrapping the strip around a cylinder Notation Here yz represents y and a 1 in the upper left cell corresponds to the threevariable minterm x y Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 633 yz yz yz yz NX 93AyA 3V xAyA 3EyA 3 Two adjacent cells represent two threevariable minterms which can be reduced to a twovariable term Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 733 yz yz yz yz NxAyAZVxAyAZ E Nai z Two adjacent cells represent two threevariable minterms which can be reduced to a twovariable term Note that due to wraparound the two highlighted cells are considered to be adjacent Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 833 yz yz yz yz NX xAyMN2VxAyAZV NCCMyMN3VxA 7 S gt 2 33 Ill 2 2 A 2X2 block of cells represents 4 minterms which can be reduced to a single variable Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 933 yz yz yz yz NxAyAZVxAyA2V NxAyA3VxAyAz Ill 2 R A 4X1block of cells represents 4 minterms which can be reduced to a single variable Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1033 yz yz yz yz yAzxyz NyZV93yZV AyAzxyz yZVN rmW 375 A 4X2 block of cells represents 8 minterms which can be reduced to a tautology Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1133 A block of ls is called an implicant of the expression being minimized It is a prime implicant if it is not contained in a larger block The goal is to identify the largest possible blocks in the map and cover all the 1s in the map by the fewest number of blocks In general the largest blocks are chosen since they give the most simplification A block which is the only one to cover a particular 1 is called a essential prime implicant and must be chosen By covering all blocks with prime implicants we can massage the expression into an connected set of prime implicants Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1233 Minimize xAyAz xAyAz CUNyWWNZD V CCNyWW V yZ V 93yZ V N x N y z E 7 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1333 Minimize xAyAz xAyAz CUNyWWNZD V CCNyWW V yZ V 93yZ V N x N y z E 7 First we fill in the Kmap yz yz yz yz Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1333 We can break the Kmap into three blocks yz yNZ NyXNZ Nyz yz yNZ NyXNZ Nyz yz yNZ NyNZ Nyz 56 N y Z which shows that the expression is equivalent to xVyz Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1433 Minimize 93 A y A N 2 ltlt as A lt y A lt 2 7 2 3 gt 2 S gt 33 Ill First we fill in the Kmap yz yz yz yz NX Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1533 We can break the Kmap into two blocks yz yNZ NyNZ Nyz yz yNZ NyXNZ Nyz X JIAOVZ 56Ay which shows that the expression is equivalent to 56 N 2 V N 93 N y Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1633 YZ yZ NYXNZ NYZ WX NW NX wX wIyz V wIyz EwAIz A 2X1block of cells represents 2 minterms which can be reduced to a 3variable expression Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1733 Note The top and bottom of the square are considered adjacent and so are the left and right topologically it is a torus donut Figure The folding of a 4x4 and a 8x8 square into a torus Logic 7 Minimization of Expressions Karnaughmaps and the QuineMCCluskey Method 7 p 1833 YZ yZ NYXNZ NYZ WX NW NX wX ltltwgtAltxgtAyAzgt v ltltwgtAltmgtAyAltzgtgt v NwAiEAyA2 V NwAyAz Ewv A 4X1block of cells represents 4 minterms which can be reduced to a 2variable expression Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 1933 YZ yZ NYXNZ NYZ WX NW NX wX wIyz V wIyz V wIyz V wIyz E 1 z A 2X2 block of cells represents 4 minterms which can be reduced to a 2variable expression Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2033 YZ yZ NYXNZ NYZ WX NW NX wX wcyz ltwltmgtAyAltzgtgt ltltwgtAltmgtAyAltzgtgt ltltwgtmAyAltzgtgt E z ltwmltygtAltzgtgt v ltwltmgtAltygtAltzgtgt v ltltwgtAltmgtAltygtAltzgtgt v ltltwgtmAltygtAltzgtgt A 2X4 block of cells represents 8 minterms which can be reduced to a 1variable expression Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2133 Minimize wcyz wcyz wAiEAyAZ V wAiEAyAZ V ltltwgtmAltygtAltzgtgt v ltltwgtAltmgtAyAltzgtgt v E 7 w A iv A N y A 2 We fill in the Kmap yZ yZ yZV MyZ WX WX WX Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2233 We can break the Kmap into three blocks yz yNz NyNz Nyz Wx Wx WNX WNx NWXNX NWXNX Nwx Nwx wIIIy NIEz which shows that the expression is equivalent to W WNX NWXNX NWX N y N z wAxAyVxAZVyAZ Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2333 Each additional variable doubles either the number of columns or the number of rows alternating in the associated Kmap Clearly as the number of variables grows this procedure becomes more complicated and it gets more difficult to see what the best splitting of the Kmap is Note In most cases there are more than one minimal expression ie the blocking step is nonunique Note The use of Kmaps relies manual visual inspection This approach is difficult to automate As the use of Kmaps gets cumbersome for large gt 4 number of variables we now look at a method which can be automated the QuineMcCuskey Method Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2433 C The Quine McCluskey method is an automatable twostage scheme for minimizing an expression gt In the first stage we figure out what terms can be combined to form terms with fewer variables gt In the second stage we figure out which of the fewervariable terms are needed to cover the expression A couple of examples will shed some light on the issue Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2533 Minimize ivyAz v EyZ V NmAyAZ V WMWWW V ltltmgtAltygtAltzgtgt E 727 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2633 Minimize acyAz cyz cyz ltltxgtAltygtAzgt v ltltxgtAltygtAltzgtgt 2 i For notational convenience we code the expressions using binary notation 1s for the variables and Us for not39ed variables here Expression Binary Code x y z 111 x N y z 101 N x y z 011 N 93 N y Z 001 6 A y A 2 000 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2633 C The Hamming distance between two binary expressions is the num ber of differing bits In our case codes with Hamming distance 1 correspond to logic expressions which can be simplified We build a table in the following way Code Combo Code2 11 111 12 11 21 101 13 11 31 011 24 01 41 001 34 01 51 000 45 00 1Used in Step 1 Step 1 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2733 C The Hamming distance between two binary expressions is the num ber of differing bits In our case codes with Hamming distance 1 correspond to logic expressions which can be simplified We build a table in the following way Code Combo Code2 Combo Code3 11 111 122 11 1234 1 21 101 132 11 31 011 242 01 41 001 342 01 51 000 45 00 1234 can be formed in two ways 1234 and 1324 The result is the same therefore we consider 12 13 24 and 34 used in the second step Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2733 C At this point not more combinations can be made we set up a table with columns for all the initial combinations and one row for each terminal unused combination and mark the common elements 1 2 3 4 5 1234 V V V V 45 V V Next we select the smallest number of combinations which cover all the initial singlets in this case we need both 1234 and 45 We have Tag Binary Logical 1234 1 z 45 00 N 56 A N y Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2833 C Therefore we have acyAz acyz cyz AyAZ V AyAZ ZVNXNY Comment In this case the minimal expression is unique that is not true in general there may be several equivalent minimal expressions Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 2933 Minimize wcyZ wcyz wcyz wcyz ltltwgtmAltygtAzgt v ltltwgtAltxgtAyAzgt v w c yz E 7 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 3033 Minimize wszAyAz v wCyZ v NwAvAyAZ V wAvAyAZ V NwAAyAZ V NwAvAyAZ V w A N iv A 9 A 2 E 77 Expression Binary Code waAyA 1110 w xyz 1011 wxyz 0111 w N x y N 1010 N w x N y z 0101 N w N x y z 0011 N w N x N y Z 0001 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 3033 C Singlet Code Pair Code Quad Code 1gtlt 1110 14 110 3567 01 2gtlt 1011 24 101 3gtlt 0111 26 011 4gtlt 1010 35gtlt 011 5gtlt 0101 36gtlt 011 6gtlt 0011 57gtlt 001 7gtlt 0001 67gtlt 001 1 2 3 4 5 6 7 3567 X X X X 14 X X 24 X X 26 X X Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 3133 1 2 3 4 5 5 7 O1 X X X X 110 X X 101 X X 011 X X The top two expressions are needed to cover 3 and Additionally one of the two bottom ones are needed Therefore w y N Z w y N Z V V wIy V V xyz are two possible minimal eXpressions equivalent to the initial one Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 3233 Homework Homework 3 Due Friday 9292006 Final Version Use both Kmaps and the Quine McCluskey method to minimize the following two expressions 1 IIINZJAZ V IIIANZJANZ V NAyZ V NAyZ V ltltxgtAltygtAltzgtgt 2 i 2 wAchyAz wcyz wcyz wcyz ltwltmgtAltygtAltzgtgtgt v ltltwgtmAyAzgtgt v ltltwgtmAyAltzgtgtgt v ltltwgtmAltygtAltzgtgtgt v ltltwgtmAltygtAzgtgt v ltltwgtAltmgtAyAltzgtgtgt v w A N iv A 9 A N Z 777 Logic i Minimization of Expressions Karnaugh maps and the Quine MCCluskey Method 7 p 3333

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