LECTURE # 3 Laws of Logic

Transcription

1 APPLYING LAWS O LOGIC Using law of logic, simlify the statement form [~(~ )] Solution: [~(~ )] [~(~) (~)] LECURE # 3 Laws of Logic [ (~)] [ ] (~) (~) Which is the simlified statement form. EXAMPLE Using Laws of Logic, verify the logical euivalence ~ (~ ) ( ) DeMorgan s Law Double Negative Law Associative Law for Indemotent Law ~(~ ) ( ) (~(~) ~) ( ) DeMorgan s Law ( ~) ( ) Double Negative Law (~ ) Distributive Law c Negation Law Identity Law SIMPLIYING A SAEMEN: You will get an A if you are hardworking and the sun shines, or you are hardworking and it rains. Rehrase the condition more simly. Solution: Let = You are hardworking = he sun shines r = It rains.he condition is then ( ) ( r) And using distributive law in reverse, ( ) ( r) ( r) Putting ( r) back into English, we can rehrase the given sentence as You will get an A if you are hardworking and the sun shines or it rains. EXERCISE: Use Logical Euivalence to rewrite each of the following sentences more simly. 1.It is not true that I am tired and you are smart. {I am not tired or you are not smart.} 2.It is not true that I am tired or you are smart. {I am not tired and you are not smart.} 3.I forgot my en or my bag and I forgot my en or my glasses. {I forgot my en or I forgot my bag and glasses. 4.It is raining and I have forgotten my umbrella, or it is raining and I have forgotten my hat. {It is raining and I have forgotten my umbrella or my hat.} CONDIIONAL SAEMENS: Introduction Consider the statement: "If you earn an A in Math, then I'll buy you a comuter." his statement is made u of two simler statements: : "You earn an A in Math," and : "I will buy you a comuter." he original statement is then saying :

2 if is true, then is true, or, more simly, if, then. We can also hrase this as imlies, and we write. CONDIIONAL SAEMENS OR IMPLICAIONS: If and are statement variables, the conditional of by is If then or imlies and is denoted. It is false when is true and is false; otherwise it is true. he arrow " " is the conditional oerator, and in the statement is called the hyothesis (or antecedent) and is called the conclusion (or conseuent). RUH ABLE: PRACICE WIH CONDIIONAL SAEMENS: Determine the truth value of each of the following conditional statements: 1. If 1 = 1, then 3 = 3. RUE 2. If 1 = 1, then 2 = 3. ALSE 3. If 1 = 0, then 3 = 3. RUE 4. If 1 = 2, then 2 = 3. RUE 5. If 1 = 1,then 1 = 2 and 2 = 3. ALSE 6. If 1 = 3 or 1 = 2 then 3 = 3. RUE ALERNAIVE WAYS O EXPRESSING IMPLICAIONS: he imlication could be exressed in many alternative ways as: if then not unless imlies follows from if, if only if whenever is sufficient for is necessary for EXERCISE: Write the following statements in the form if, then in English. a)your guarantee is good only if you bought your CD less than 90 days ago. If your guarantee is good, then you must have bought your CD layer less than 90 days ago. b)o get tenure as a rofessor, it is sufficient to be world-famous. If you are world-famous, then you will get tenure as a rofessor. c)hat you get the job imlies that you have the best credentials. If you get the job, then you have the best credentials. d)it is necessary to walk 8 miles to get to the to of the Peak. If you get to the to of the eak, then you must have walked 8 miles. RANSLAING ENGLISH SENENCES O SYMBOLS: Let and be roositions: = you get an A on the final exam = you do every exercise in this book

3 r = you get an A in this class Write the following roositions using,,and r and logical connectives. 1.o get an A in this class it is necessary for you to get an A on the final. SOLUION r 2.You do every exercise in this book; You get an A on the final, imlies, you get an A in the class. SOLUION r 3. Getting an A on the final and doing every exercise in this book is sufficient or getting an A in this class. SOLUION r RANSLAING SYMBOLIC PROPOSIIONS O ENGLISH: Let,, and r be the roositions: = you have the flu = you miss the final exam r = you ass the course Exress the following roositions as an English sentence. 1. If you have flu, then you will miss the final exam.2.~ r If you don t miss the final exam, you will ass the course.3.~ ~ r If you neither have flu nor miss the final exam, then you will ass the course. HIERARCHY O OPERAIONS OR LOGICAL CONNECIVES ~ (negation) (conjunction), (disjunction) (conditional) Construct a truth table for the statement form ~ ~ ~ ~ ~ ~ ~

4 Construct a truth table for the statement form ( ) (~ r) ( ) (~ r) ~ r ~ r LOGICAL EQUIVALENCE INVOLVING IMPLICAION Use truth table to show ~ ~ same truth values ~ ~ ~ ~

5 Hence the given two exressions are euivalent. IMPLICAION LAW ~ ~ ~ same truth values NEGAION O A CONDIIONAL SAEMEN: Since ~ therefore ~ ( ) ~ (~ ) ~ (~ ) (~ ) by De Morgan s law ~ by the Double Negative law hus the negation of if then is logically euivalent to and not. Accordingly, the negation of an if-then statement does not start with the word if. EXAMPLES Write negations of each of the following statements: 1.If Ali lives in Pakistan then he lives in Lahore. 2.If my car is in the reair sho, then I cannot get to class. 3.If x is rime then x is odd or x is 2. 4.If n is divisible by 6, then n is divisible by 2 and n is divisible by 3. SOLUIONS: 1. Ali lives in Pakistan and he does not live in Lahore. 2. My car is in the reair sho and I can get to class. 3.x is rime but x is not odd and x is not 2. 4.n is divisible by 6 but n is not divisible by 2 or by 3. INVERSE O A CONDIIONAL SAEMEN: he inverse of the conditional statement is ~ ~ A conditional and its inverse are not euivalent as could be seen from the truth table. ~ ~ ~ ~

6 different truth values in rows 2 and 3 WRIING INVERSE: 1. If today is riday, then = 5. If today is not riday, then If it snows today, I will ski tomorrow. If it does not snow today I will not ski tomorrow. 3. If P is a suare, then P is a rectangle. If P is not a suare then P is not a rectangle. 4. If my car is in the reair sho, then I cannot get to class. If my car is not in the reair sho, then I shall get to the class. CONVERSE O A CONDIIONAL SAEMEN: he converse of the conditional statement is A conditional and its converse are not euivalent. i.e., is not a commutative oerator. not the same WRIING CONVERSE: 1.If today is riday, then = 5. If = 5, then today is riday. 2.If it snows today, I will ski tomorrow. I will ski tomorrow only if it snows today. 3. If P is a suare, then P is a rectangle. If P is a rectangle then P is a suare. 4. If my car is in the reair sho, then I cannot get to class. If I cannot get to the class, then my car is in the reair sho. CONRAPOSIIVE O A CONDIIONAL SAEMEN: he contraositive of the conditional statement is~ ~ A conditional and its contraositive are euivalent. Symbolically, ~ ~ 1.If today is riday, then = 5. If , then today is not riday. 2.If it snows today, I will ski tomorrow. I will not ski tomorrow only if it does not snow today. 3. If P is a suare, then P is a rectangle. If P is not a rectangle then P is not a suare. 4. If my car is in the reair sho, then I cannot get to class. If I get to the class, then my car is not in the reair sho.