1 The Foundations. 1.1 Logic. A proposition is a declarative sentence that is either true or false, but not both.

Transcription

1 he oundations. Logic Propositions are building blocks of logic. A proposition is a declarative sentence that is either true or false, but not both. Example. Declarative sentences.. Ottawa is the capital Canada. 2. oronto is the capital Canada = =. 5. George Bush is the president. All 5 sentences are declarative, so all are propositions. Sentences and 3 are true, the rest are false. Why? Remark: Correctness may depend on time, system, place, etc. Example 2. Non-declarative sentences not propositions!. What is your name? 2. Stand up! 3. 2+x=2. 4. x+y=z. Sentences and 2 are not declarative; for 3 and 4, a definitive judgement cannot be made. So, none of them are propositions. Propositions are denoted by letters like p, q, r, s,. If a proposition is true, the truth value of this proposition is denoted by or by ; if the proposition is not true, then it is denoted by or by (zero). he area of logic that deals with propositions is called propositional calculus/logic. Propositions, constructed by combining one or more existing propositions using logical operators, are called compound proposition.

2 Definition Let p be a proposition. he statement It is not the case that p is another proposition, called the negation of p. he negation of p is denoted by p. he proposition p is read not p. Example 3. ind the negation of the proposition oday is riday. and express it in simple English. Solution: he negation is It is not the case that today is riday. Or simply oday is not riday. Remark: Strictly speaking, sentences involving variable times such as those in Example 3 are not propositions, unless a fixed time and further a fixed position/place is assumed. A truth table displays the relationships between the truth values of propositions. ABLE he ruth able for the Negation of a Proposition. p p he negation p of a proposition p can also be considered the result of the operation of the negation operator on a proposition p. he logical operators that are used to form new propositions from two or more existing propositions are called connectives. 2

3 Definition 2 Let p and q be a proposition. he proposition p and q, denoted by p ^ q, is the proposition that is true when both p and q are true and is false otherwise. he proposition p ^ q is called the conjunction of p and q. ABLE 2 he ruth able for the Conjunction of two Propositions. p q p ^ q Example 4. ind the conjunction of the proposition p and q where p is the proposition oday is riday. And q is the proposition It is raining today. Solution: p ^ q is the proposition oday is riday and it is raining today. Definition 3 Let p and q be propositions. he proposition p or q, denoted by p v q, is the proposition that is false when both p and q are false and is true otherwise. he proposition p v q is called the disjunction of p or q. ABLE 3 he ruth able for the Disjunction of two Propositions. p q p v q 3

4 Example5. ind the disjunction of the proposition p and q where p is the proposition oday is riday. And q is the proposition It is raining today. Solution: p v q is the proposition oday is riday or it is raining today. Definition 4 Let p and q be propositions. he exclusive or of p and q, denoted by p q, is the proposition that is true when exactly one of p and q is true and is false otherwise. ABLE 4 he ruth able for the Exclusive Or of two Propositions. p q p q Implications ABLE 5 he ruth able for the Implication p q. p q p q Definition 5 Let p and q be propositions. he implication p q is the proposition that is only false when p is true and q is false, and true otherwise. In this implication p is called the hypothesis (or premise) and q is called the conclusion (or consequence). An implication is sometimes called a conditional statement. If p is false, then p q is always true. If a politician is not elected, you cannot say, he has broken his campaign pledge and he is an untruthful person. You can say it only if he is elected but broke his campaign pledge. Remark: he way we have defined implications is more general than the meaning attached to language. he implication 4

5 If today is riday, then 2+3=6. Is false only if today is riday, is true all the other days, even though 2+3=6 is false. here are many different ways to express implication p q : if p, then q p implies q if p, q p only if q p is sufficient for q, a sufficient condition for q is p q if p q whenever p q when p q is necessary for p a necessary condition for p is q q follows from p q unless p or comprehension, the truth value of a conditional statement can be compared to a contract or a promise. Remark: p only if q q p, if q then p. Note that p only if q says p cannot be true when q is not true. q unless not p = p q. Check the truth value. However the if-then construction used in many programming languages is different from that used in logic. In the statement if p, then S, S is executed if p is true, but S is not executed if p is false. Example 6. What is the value of the variable x after the statement If 2+2=4 then x:=x+ If x= before this statement is encountered? Solution:.. int x=;. printf( the value before: %d\n, x); if (2+2==4) then x=x+; printf( the value after: %d\n, x);.. 5

6 CONVERSE, CONRAPOSIIVE AND INVERSE here are some related implications that can be formed from p q. he proposition q p is called the converse of p q. he contrapositive of p q is the proposition q p. he proposition p q is called the inverse of p q. Evidently the contrapositive, q p, of an implication p q has the same truth value as p q and therefore they are equivalent. Example 7. What are the contrapositive, the converse and the inverse of the implication he home team wins whenever it is raining.? Solution: q whenever p is equivalent to the implication p q If p, then q.. So p is It is raining. And q is he home team wins. So the contrapositive, q p: If home team does not win, then it is not raining. he converse, q p: If home team wins, then it is raining. he inverse, p q: If it is not raining, then the home team does not win. Definition 6 Let p and q be propositions. he biconditional p q is the proposition that is true when p and q have the same truth values, and is false otherwise. Clearly p q is equivalent to (p q) (q p), in verbal expression p if and only if q, p is necessary and sufficient for q, if p then q, conversely 6

7 ABLE 6 he ruth able for the Biconditional p q. p q p q Example 8. Let p be the statement You can take the flight and q the statement You buy a ticket. hen p q is the statement You can take the flight if and only if you buy a ticket. Remark: precision in essential in math and logic, but it may not be the case in language. he oundations PRECEDENCE O LOGICAL OPERAORS Using logical operators, compound operators can be constructed. So the precedence of logical operators does matters. ABLE 7 Precedence of Logical Operators. Operator Precedence herefore p q=( p) q, p q r=(p q) r and p q r = (p q) r. 7

8 RANSLAING ENGLISH SENENCES here are many reasons to translate English/human language sentences into expressions involving propositional variables and logical connectives. In particular, English/human language is often ambiguous. ranslating sentences into logical expressions removes the ambiguity. Example 9. How can the English sentence You can access the Internet from campus only if you are a computer science major or you are not a freshmen. into a logical expression? Solution: we have y only if x and this is actually y is sufficient to x and again this means if y then x. Notice that here x is a compound. Let a be You can access the Internet from campus., c be You are a computer science major and f be You are a freshmen. hen the sentence is a only if c or f, in another word if a then c or f and this is a (c f ). Example. How can the English sentence You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 6 years old. into a logical expression? Solution: this is obviously y if x. So it is equal to if x then y. However the x here is again a compound. Rewriting you are under 4 feet tall unless you are older than 6 years old as you are under 4 feet tall with the exception that you are older than 6 years, and this is you are under 4 feet tall and it is not the case that you are older than 6 years everything is quite clear. cannot ride can ride Set of people under 4 feet tall People not over 6 years people over 6 years 8

9 Let r be You can ride the roller coaster., s be you are under 4 feet tall (s for short), o be you are older than 6 years. hen the sentence is (s o ) r. In another word, You can ride the roller coaster if you are not under 4 feet tall or you are older than 6 years old. he rest cannot ride(short and not old cannot ride). Solution2: all(not short) or old can ride and this is ( s o ) r. So the negation of ( s o ) r is cannot ride only if it is not the case that tall(not short) or old and this means ( s o ) r. Now looking at the truth table s o s o ( s o ) s o we see that ( s o )= s o. hus the original sentence in logical expression is : (s o) r, in words short and not old cannot ride. SYSEM SPECIICAIONS ranslating sentences in natural language into logical expressions is an essential part of specifying both hardware and software systems. Purpose is to take requirements in natural languages and to produce precise and unambiguous (clear) specifications that can be used as the basis for system development. Example. Express the specification he automated reply cannot be sent when the file system is full. using logical connectives. Solution: this is again y if x and this is if x then y. 9

10 System specifications should not contain conflicting requirements. Otherwise there would be no way to develop a system that satisfies all the specifications. Consequently, propositional expressions representing these specifications need to be consistent. hat is, there must be an assignment of truth values to the variables in the expressions that makes all the expressions true. Example 2. Determine whether these system specifications are consistent: he diagnostic message is stored in the buffer or it is retransmitted. he diagnostic message is not stored in the buffer. If the diagnostic message is not stored in the buffer, then it is retransmitted. Solution: to check the consistency express them using logical expressions and check the truth table if all three propositions can be true for the variable propositions. Let s: he diagnostic message is stored in the buffer. r: he diagnostic message is retransmitted. hen the specifications are: s r, s, s r. Checking the truth table s r s r s s r we see that the assignment of the variables s= and r= makes all three expressions true. So the system is consistent. Example 3. Do the system specifications in Example 2 remain consistent if the specification he diagnostic message is not retransmitted. is added? Solution: adding one implication to the truth table of Example 2 and looking at the truth values we see that the system specifications are inconsistent.

11 BOOLEAN SEARCHES Logical connectives are used extensively in searches of large collections of information, such as indexes of Web pages. Since these searches use the techniques from propositional logic, they are called Boolean searches. In Boolean searches AND is used to records match both terms, OR is used to match one or both of search terms, NO is used to exclude a particular term. Example 4. Webpage searching. ind universities in New Mexico. Solution: using twice AND can solve the problem. LOGIC AND BI OPERAIONS Computers represent information using bits. A bit has two possible values: and. A bit can also be used to represent a truth value: true or false. A variable is called a Boolean variable if its value is either true or false. A boolean variable can be represented using a bit. Computer bit operations corresponds logical connectives. ABLE 8 able for the Bit Operators OR, AND, and XOR. x y x y x y x y Definition 7 A bit string is a sequence of zero or more bits. he length of this string is the number of bits in the string. Example 7. is a bit string of length nine.

12 Example 8. ind bitwise OR, bitwise AND, and bitwise XOR of the bit strings and. Solution: ========== bitwise OR bitwise AND bitwise XOR LOGIC PUZZLES Puzzles that can be solved using logical reasoning are called logic puzzles. Example 5. An island has two kinds of habitants, knights, who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. What are A and B if A says B is knight and B says he two of us are opposite types? Solution: If A is true then according to A s statement B is knight, B should be true too, this means A B. If B is true then A B should be true and this means B A B. able for Logic puzzles A B A B B A B If A is false then A said B is knight, B should be false too, this means A B. If B is false, then what he said A B should be false too and this means B (A B). 2

13 able for Logic puzzles 2 A B A B B (A B) rom the tables we see that able 2 is consistent and this means A and B are both knaves. 3