22C:19 Discrete Math. So. What is it? Why discrete math? Fall 2009 Hantao Zhang
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1 22C:19 Discrete Math Fall 2009 Hantao Zhang So. What is it? Discrete mathematics is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity (wikipedia) In other words, dealing with integer things (sets, logic, proofs, etc.) and not continuous things (calculus, functions, etc.) Why discrete math? It forms the basis for computer science: Sequences Digital logic (how computers compute) Algorithms Assuring computer correctness Probability and gambling (really!) Etc. Like how regular math forms the basis for science 1
2 Oldest and simplest Applications: Computer programs Computer addition Logic problems Sudoku Boolean logic 4 Boolean propositions A proposition is a statement that can be either true or false The sky is blue Today is sunny x==y Not propositions: Are you Bob? x:=7 5 Boolean variables We use Boolean variables to refer to propositions Usually are lower case letters starting with p (i.e. p, q, r, s, etc.) A Boolean variable can have one of two values true (T) or false (F) A proposition can be A single variable: p A formula of multiple variables: p (q r) 6 2
3 Logical Operators About a dozen logical operators Similar to algebraic operators + * - / In the following examples, p =: Today is Fid Friday q =: Today is my birthday =: reads as represents or means p is the syntax and Today is Friday is the semantics. 7 Logical operators: Not A not operation switches (negates) the truth value p p Symbol: or ~ T F In C++ and Java, F T the operand is! p =: Today is not Friday 8 Logical operators: And An and operation is true if both operands are true Symbol: p q p q It s like the A in And T T T In C++ and Java, T F F the operand is && F T F p q =: Today is Friday and F F F today is my birthday 9 3
4 Logical operators: Or Anor operation is true if either operands are true Symbol: p q p q In C++ and Java, T T T the operand is T F T F T T p q =: Today is Friday or F F F today is my birthday (or possibly both) 10 Logical operators: Exclusive Or An exclusive or operation is true if one of the operands are true, but false if both are true Symbol: p q p q Often called XOR T T F p q (p q) (p q) T F T In Java, the operand is ^ F T T (in C++, it s bitwise exclusive or) F F F p q =: Today is Friday or today is my birthday, but not both 11 Inclusive Or versus Exclusive Or Do these sentences mean inclusive or exclusive or? Experience with C++ or Java is required Lunch includes soup or salad To enter the country, you need a passport or a driver s license Publish or perish 12 4
5 Logical operators: Nand and Nor The negation of And and Or, respectively Symbols: and, respectively Nand: p q (p q) Nor: p q (p q) p q p q p q p q p q T T T T F F T F F T T F F T F T T F F F F F T T 13 Logical operators: Conditional Aconditional (also called implication) means if p then q p q p q p q Symbol: T T T T p q =: If today is Friday, then today is my birthday p q p q T F F F F T T T F F T T the antecedent the consequence 14 Logical operators: Conditional Let p = I am elected and q = I will lower taxes I state: p q =: If I p q p q am elected, then I T T T will lower taxes T F F F T T Consider all possibilities F F T Note that if p is false, then the conditional is true regardless of whether q is true or false 15 5
6 Logical operators: Conditional Conditional Inverse Converse Contrapositive p q p q p q p q q p q p T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T 16 Logical operators: Conditional Alternate ways of stating a conditional: p implies q Ifp, q p is sufficient for q q if p q whenever p q is necessary for p p only if q This one is confusing 17 Logical operators: Bi-conditional A bi-conditional means p if and only if q Symbol: Alternatively, it means (if p then q) and (if q then p) p q p q q p Note that a bi-conditional has the opposite truth values of the exclusive or p q p q T T T T F F F T F F F T 18 6
7 Logical operators: Bi-conditional Let p =: You take this class and q =: You get a grade p q p q Then p q means You take this class if T T T and only if you get a grade Alternatively, it means If you take this class, then T F F F T F F F T you get a grade and if you get a grade then you take (took) this class 19 Boolean operators summary not not and or xor nand nor conditional biconditional p q p q p q p q p q p q p q p q p q T T F F T T F F F T T T F F T F T T T F F F F T T F F T T T F T F F F T T F F F T T T T Learn what they mean, don t just memorize the table! 20 Precedence of operators Just as in algebra, boolean operators have precedence 4+3*2 = 4+(3*2), not (4+3)*2 Precedence order (from highest to lowest): The first three are the most important This means that p q r s t yields: (p (q ( r))) (s t) Not is always performed before any other operation 21 7
8 Translating English Sentences Problem: p = It is below freezing q = It is snowing It is below freezing and it is snowing p q It is below freezing but not snowing p q It is not below freezing and it is not snowing p q It is either snowing or below freezing (or both) p q If it is below freezing, it is also snowing p q It is either below freezing or it is snowing, (p q) (p q) but it is not snowing if it is below freezing That it is below freezing is necessary and p q sufficient for it to be snowing 22 Translation Example I have neither given nor received help on this exam Rephrased: I have not given nor received Let p =: I have given help on this exam Let q =: I have received help on this exam Translation is: p q p q p p q T T F F T F F T F T T F F F T F 23 Translation Example What they mean is I have not given and I have not received help on this exam Or I have not (given nor received) help on this exam p q p q p q T T F F T F F F F T F F F F T T The problem: has a higher precedence than in Boolean logic, but not always in English Also, neither is vague 24 8
9 Boolean Searches (21 OR 22) AND university of iowa AND computer science Note that Google requires you to capitalize Boolean operators Google defaults to AND; many others do not So the AND s were optional XOR doesn t work 25 Bit Operations Boolean values can be represented as 1 (true) and 0 (false) A bit string is a series of Boolean values is eight Boolean values in one string We can then do operations on these Boolean strings Each column is its own Boolean operation Bit Operations 2 Evaluate the following ( ) = (11011) =
10 Consider the following: int p = 11; int q = 20; if ( p && q ) { } if ( p & q ) { } Bitwise Boolean And operation: && vs. & in C/C++ In C/C++, any value other than 0 is true The binary for the integer 11 is The binary for the integer 20 is Notice the double ampersand this is a Boolean operation As p and q are both true, this is true Noticethesingleampersand thisisa bitwise operation This evaluates to zero (false)! 28 && vs. & in C/C++ Note that Java does not have this feature Ifp and q are int: p & q is bitwise p && q will not compile Ifp and q are boolean: Both p & q and p && q will be a Boolean operation The same holds true for the or operators ( and ) in both Java and C/C++ 29 Tautology and Contradiction A tautology is a statement that is always true p p will always be true (Negation Law) A contradiction is a statement that is always false p p will always be false (Negation Law) p p p p p T T F F T F 30 10
11 What is a Fallacy? A fallacy is an error in an argument. An argument consists of one or more premises and one conclusion. Both the premises (p 1, p 2,,p n ) and the conclusion (q) are statements (i.e., true or false). If(p 1,p 2,,p n q) is not a tautology, then a fallacy occurs. Ifp i is false, then a fallacy occurs. 31 Logical Equivalence A logical equivalence means that the two sides always have the same truth values Symbol is or We ll use, so as not to confuse it with the biconditional 32 Logical Equivalences of And p T p Identity law p T p T T T T F T F p F F Domination law p F p F T F F F F F 33 11
12 Logical Equivalences of And p p p Idempotent law p p p p T T T F F F p q q p Commutative law p q p q q p T T T T T F F F F T F F F F F F 34 Logical Equivalences of And (p q) r p (q r) Associative law p q r p q (p q) r q r p (q r) T T T T T T T T T F T F F F T F T F F F F T F F F F F F F T T F F T F F T F F F F F F F T F F F F F F F F F F F 35 Logical Equivalences of Or p T T p F p p p p p q q p (p q) r p (q r) Identity law Domination law Idempotent law Commutative law Associative law 36 12
13 Associative Laws of And and Or (p q) r p q r (p q) r p q r Similar to (3+4)+5 = Only works if ALL the operators are the same! 37 Logical Equivalences of Not ( p) p p p T p p F Double negation law Negation law Negation law 38 DeMorgan s Law Probably the most important logical equivalence To negate p q (orp q), you flip the sign, and negate BOTH p and q Thus, (p q) p q Thus, (p q) p q pq p q p q (p q) p q p q (p q) p q TT F F T F F T F F TF F T F T T T F F FT T F F T T T F F FF T T F T T F T T 39 13
14 Yet more equivalences Distributive: p (q r) (p q) (p r) p (q r) (p q) (p r) Absorption p (p q) p p (p q) p 40 How to prove two propositions are equivalent? Two methods: Using truth tables Not good for long formulae Using the logical equivalences Can be automated Example: show that: ( p r) ( q r) ( p q) r 41 Using Truth Tables ( p r) ( q r) ( p q) r p q r p r q r (p r) (q r) p q (p q) r T T T T T T T T T T F F F F T F T F T T T T F T T F F F T T F T F T T T T T F T F T F T F T F T F F T T T T F T F F F T T T F T 42 14
15 Using Logical Equivalences ( p r) ( q r) ( p q) r Original statement p r) ( q r) ( p pq ) qr p q ( Definition of implication ( p DeMorgan s r ) ( q Law r ) ( p ( p q ) r p q Associativity p r q of ror ( p rq) ( r q r) p r q r Re-arranging p q r r p q r p q r r p r rq r Idempotent Law 43 Logical Thinking At a trial: Bill says: Sue is guilty and Fred is innocent. Sue says: If Bill is guilty, then so is Fred. Fred says: I am innocent, but at least one of the others is guilty. Let b =: Bill is innocent, f = Fred is innocent, and s = Sue is innocent Statements are: s f b f f ( b s) 44 Can all of their statements be true? Show: ( s f) ( b f) (f ( b s)) b f s b f s s f b f b s f ( b s) T T T F F F F T F F T T F F F T T T T T T F T F T F F T F F T F F F T T F T T F F T T T F F F F T T F T F T F T T F T T F F T T T F F T T F F F F T T T F T T F 45 15
16 Are all of their statements true? Show values for s, b, and f such that the equation is true ( s f ) ( b f ) ( f ( b s)) T ( s f ) ( b f ) ( f ( b s)) T s f ( b f ) f ( b s) T s f f ( b f ) ( b s) T s f ( b f ) ( b s) T f ( b f ) s ( s b) T f ( b f ) s T ( f ( b f )) s T (( f b) ( f f )) s T (( f b) F ) s T ( f b) s T f b s T Original statement Definition of implication Associativity of AND Re-arranging Idempotent t law Re-arranging Absorption law Re-arranging Distributive law Negation law Domination law Associativity of AND 46 What if it weren t possible to assign such values to s, b, and f? ( s f ) ( b f ) ( f ( b s)) s = T ( s f ) ( b f ) ( f ( b s)) s = T ( f b) s s = T f b s s = T f b F = T f F = T F = T Original statement Definition of implication... (same as previous slide) Domination law Re-arranging Negation law Domination law Domination law Contradiction! 47 Functional completeness All the extended operators have equivalences using only the 3 basic operators (and, or, not) The extended operators: nand, nor, xor, conditional, bi-conditional Given a limited set of operators, can you write an equivalence of the 3 basic operators? If so, then that group of operators is functionally complete 48 16
17 Functional completeness of NAND Show that (NAND) is functionally complete Equivalence of NOT: p p p (p p) p Equivalence of NAND (p) p Idempotent law 49 Functional completeness of NAND Equivalence of AND: p q (p q) Definition of nand p p How to do a not using nands (p q) (p q) Negation of (p q) Equivalence of OR: p q ( p q) DeMorgan s equivalence of OR As we can do AND and OR with NANDs, we can thus do ORs with NANDs Thus, NAND is functionally complete 50 17
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