Boolean Logic CSAPMA 202 Spring 2005 Rosen section

Boolean Logic CS/APMA 202, Spring 2005 Rosen, section 1. 1 Aaron Bloomfield 1

Quick survey n How familiar are you with Boolean logic? a) b) c) d) A lot Some A little Boolean what? 2

Boolean propositions A proposition is a statement that can be either true or false n n n “The sky is blue” “I is a English major” “x == y” Not propositions: n n “Are you Bob? ” “x = 7” 3

Boolean variables We use Boolean variables to refer to propositions n n 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… n n A single variable: p An operation of multiple variables: p (q r) 4

Introduction to Logical Operators About a dozen logical operators n Similar to algebraic operators + * - / In the following examples, n p = “Today is Friday” n q = “Today is my birthday” 5

Logical operators: Not A not operation switches (negates) the truth value Symbol: or ~ In C++ and Java, the operand is ! p T F p F T p = “Today is not Friday” 6

Logical operators: And An and operation is true if both operands are true Symbol: n It’s like the ‘A’ in And In C++ and Java, the operand is && p q = “Today is Friday and today is my birthday” p T T F F q T F p q T F F F 7

Logical operators: Or An or operation is true if either operands are true Symbol: In C++ and Java, the operand is || p q = “Today is Friday or today is my birthday (or p T T F F q T F p q T T T F possibly both)” 8

Logical operators: Exclusive Or An exclusive or operation is true if one of the operands are true, but false if both are true Symbol: Often called XOR p q ( p q ) ¬ (p q ) In Java, the operand is ^ (but not in C++) p q = “Today is Friday or today is my birthday, but not both” p T T F F q T F p q F T T F 9

Inclusive Or versus Exclusive Or Do these sentences mean inclusive or exclusive or? n n 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 10

Logical operators: Nand Nor The negation of And and Or, respectively Symbols: | and ↓, respectively n n Nand: p|q ¬(p q) Nor: p↓q ¬(p q) p T T F F q T F p q T T F T F F p| q F T T T p q F F F T 11

Logical operators: Conditional 1 A conditional means “if p then q” Symbol: p q = “If today is Friday, then today is my birthday” p→q=¬p q the antecedent p T T F F q T F p q T F T T the consequence 12

Logical operators: Conditional 2 Let p = “I am elected” and q = “I will lower taxes” I state: p q = “If I am elected, then I will lower taxes” Consider all possibilities p T T F F q T F p q T F T T Note that if p is false, then the conditional is true regardless of whether q is true or false 13

Quick survey n Does truth table of the conditional make sense? a) b) c) d) Yes, already Mostly Not really Huh? I was sleeping 14

Logical operators: Conditional 3 p T T F F q p q T F F T T Conditional Inverse Contrapositive p q T F T T p q T T F T q p T T F T q p T F T T 15

Logical operators: Conditional 4 Alternate ways of stating a conditional: n n n n p implies q If p, q p only if q p is sufficient for q q if p q whenever p q is necessary for p p only if q I don’t like this one See Rosen, p. 6, for a full list 16

Logical operators: Bi-conditional 1 A bi-conditional means “p if and only if q” Symbol: p q Alternatively, it means T T “(if p then q) and T F (if q then p)” F T Note that a bi-conditional has the opposite truth values F F of the exclusive or p q T F F T 18

Logical operators: Bi-conditional 2 Let p = “You take this class” and q = “You get a grade” p q Then p q means T T T “You take this class if and only if you get a T F F grade” F T F Alternatively, it means “If F F T you take this class, then you get a grade and if you get a grade then you take (took) this class” 19

Boolean operators summary not and p q p q T T F F T F T T F F T or p q xor nand nor conditional biconditional p q p| q p q p q T F F F T T F T F F F T T Learn what they memorize the table! mean, don’t just 20

Quick survey n I understand all the Boolean operators a) b) c) d) Absolutely! With a little review after class, I’ll have it down Can we go over it again? 17 21

Precedence of operators Just as in algebra, operators have precedence n 4+3*2 = 4+(3*2), not (4+3)*2 Precedence order (from highest to lowest): ¬ →↔ n 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 22

Translating English Sentences Question 7 from Rosen, p. 16 n n p = “It is below freezing” q = “It is snowing” p q 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 It is either snowing or below freezing (or both) p q p→q If it is below freezing, it is also snowing (p q) (p→¬q) It is either below freezing or it is snowing, 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 23

Translation Example 2 Heard on the radio: n n A study showed that there was a correlation between the more children ate dinners with their families and lower rate of substance abuse by those children Announcer conclusions: If children eat more meals with their family, they will have lower substance abuse If they have a higher substance abuse rate, then they did not eat more meals with their family 24

Translation Example 2 Let p = “Child eats more meals with family” Let q = “Child has less substance abuse Announcer conclusions: n If children eat more meals with their family, they will have lower substance abuse p q n If they have a higher substance abuse rate, then they did not eat more meals with their family q p Note that p q and q p are logically equivalent 25

Translation Example 2 Let p = “Child eats more meals with family” Let q = “Child has less substance abuse” p T T F F q T F result T ? ? T conclusion T F T T 26

Translation Example 3 “I have neither given nor received help on this exam” n n n 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 T q T p F p q F T F T T F F F T F 27

Translation Example 3 What they mean is “I have not given and I have not received help on this exam” n Or “I have not (given nor received) help on this exam” p T T F q T F T p q F F F T (p q ) F F F T The problem: has a higher precedence than in Boolean logic, but not always in English Also, “neither” is vague 28

A bit of humor: Yale vs. Harvard n Web references: http: //www. harvardsucks. org/, http: //www. yaledailynews. com/article. asp? AID=27506 29

System Specifications 1 Just like translating English sentences In order for a system specification to be valid, there must be at least ONE truth assignment that allows the propositions to be true Question 43 from Rosen, p. 10 n n p = “the message is scanned for virii” q = “the message was sent from and unknown system” The message is scanned for virii whenever the message was sent from an unknown system The message was sent from an unknown system but it was not scanned for virii It is necessary to scan the message for virii whenever it was sent from an unknown system When a message is not sent from an unknown system, it is not scanned for virii q→p q ¬ p equal q→p ¬q→¬p 30

System Specifications 2 Is this a valid system? n n p = “the message is scanned for virii” q = “the message was sent from and unknown system” Requirements: (q→p) (q ¬p) (¬q→¬p) p q q p T T F T T T F F F T T F F T 31

Quick survey n Is this a valid system? a) b) c) d) Yes No Need more information I have no idea p q q p T T F T T T F F F T T F F T 32

System Specifications 2 The system as specified is not valid n There is no assignment of truth values that makes all the system specs true p q q p T T F T T T F F F T T F F T 33

System Specifications 2 Let’s remove one of the specifciations: n q ¬p = The message was sent from an unknown system but it was not scanned for virii Is this a valid system? n Requirements: (q→p) (¬q→¬p) p q q p T T F T T T F F F T T F F T 34

Boolean Searches (101 OR 202) AND bloomfield AND “computer science” Note that Google requires you to capitalize Boolean operators Google defaults to AND; many others do not 35

Bit Operations 1 Boolean values can be represented as 1 (true) and 0 (false) A bit string is a series of Boolean values n 10110100 is eight Boolean values in one string We can then do operations on these Boolean strings n Each column is its own 01011010 Boolean operation 10110100 1110 36

Bit Operations 2 Evaluate the following 11000 (01011 11011) = 11000 (11011) = 11000 01011 11011 11000 37

Quick survey n What programming languages do you know? a) b) c) d) Java only C / C++ only Java and C/C++ not (Java or C/C++) 38

&& vs. & in C/C++ Consider the following: int p = 11; int q = 20; if ( p && q ) { } if ( p & q ) { } In C/C++, any value other than 0 is true The binary for the integer 11 is 01011 The binary for the integer 20 is 10100 Notice the double ampersand – this is a Boolean operation As p and q are both true, this is true Notice the single ampersand – this is a bitwise operation 01011 Bitwise Boolean 10100 And operation: 00000 This evaluates to zero (false)! 39

&& vs. & in C/C++ Note that Java does not have this “feature” n If p and q are int: p & q is bitwise p && q will not compile n If p 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++ 40