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

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1 1 he oundations 1.1 Logic Propositions are building blocks of logic. A proposition is a declarative sentence that is either true or false, but not both. Example 1. Declarative sentences. 1. Ottawa is the capital Canada. 2. oronto is the capital Canada = = George Bush is the president. All 5 sentences are declarative, so all are propositions. Sentences 1 and 3 are true, the rest are false. Remark: Correctness may depend on time, system, place,. Example 2. Non-declarative sentences not propositions! 1. What is your name? 2. Stand up! 3. 2+x=2. 4. x+y=z. Sentences 1 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 1; if the proposition is not true, then it is denoted by or by 0(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. PD created with pdfactory trial version 1

2 Definition 1 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 1 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 PD created with pdfactory trial version

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 PD created with pdfactory trial version 3

4 Example 5. 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. PD created with pdfactory trial version 4

5 Remark: he way we have defined implications is more general than the meaning attached to language. he implication 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: if p, then q, p is sufficient for q, q when p, p implies q, p only if q, q whenever p, q is necessary for p, q follows from p.. 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+1 If x=0 before this statement is encountered? Solution:.. int x=0;. printf( the value before: %d\n, x); if (2+2==4) then x=x+1; printf( the value after: %d\n, x);.. 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. PD created with pdfactory trial version 5

6 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 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. PD created with pdfactory trial version 6