Discrete Structures Lecture Propositional Logic
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1 DEFINITION 0 EXAMPLE 1.1 EXAMPLE 1.2 EXAMPLE 1.3 EXAMPLE 1.4 EXAMPLE 2.1 EXAMPLE 2.2 EXAMPLE 2.3 EXAMPLE 2.4 A proposition is a declarative sentence that is either true or false but not both. The following is a declarative sentence. Washington D.C. is the capital of the United States of America. True. The following is a declarative sentence. Toronto is the capital of Canada. False The following is a declarative sentence. One plus one equals two = 2. True The following is a declarative sentence. Two plus two equals three = 3. False The following is not a declarative sentence because the sentence is not a statement of a fact. The sentence is a question. What time is it? The following is not a declarative sentence because the sentence is not a statement of a fact. The sentence is a command. Read this carefully. The following is not a declarative sentence because the sentence cannot be shown to be either true or false. xx + 1 = 2 The following is not a declarative sentence because the sentence cannot be shown to be either true or false. xx + yy = zz 1
2 DEFINITION 1 Let p be a proposition. The negation of p, denoted by pp (also denoted by pp), is the statement It is not the case that p. The proposition pp is read not p. The truth value of the negation of p, pp, is the opposite of the truth value of p. EXAMPLE 3 Find the negation of the proposition Michael s PC runs Linux. and express this in simple English. Solution: The negation is: It is not the case that Michael s PC runs Linux. This negation can be more simply expressed by Michael s PC does not run Linux. EXAMPLE 3.1 Find the negation of the proposition Today is Friday. and express this in simple English. Solution: The negation is: It is not the case that today is Friday. This negation can be more simply expressed by Today is not Friday. 2
3 EXAMPLE 4 Find the negation of the proposition Vandana s smartphone has at least 32GB of memory. and express this in simple English. Solution: The negation is: It is not the case that Vandana s smartphone has at least 32GB of memory. This negation can also be expressed by Vandana s smartphone does not have at least 32GB of memory. or even more simply as Vandana s smartphone has less than 32GB of memory. EXAMPLE 4.1 Find the negation of the proposition At least 10 inches of rain fell today in Miami. and express this in simple English. Solution: The negation is: It is not the case that at least 10 inches of rain fell today in Miami. This negation can be more simply expressed by Less than 10 inches of rain fell today in Miami. REMARK Table 1 displays the truth table for the negation of a proposition p. This table has a row for each of the two possible truth values of a proposition p. Each row shows the truth value of pp corresponding to the truth value of pp for this row. TABLE 1 The Truth Table for the Negation of a Proposition pp pp T F F T 3
4 REMARK The negation of a proposition can also be considered the result of the operation of the negation operator on a proposition. The negation operator constructs a new proposition from a single existing proposition. We will now introduce the logical operators that are used to form new propositions from two or more existing propositions. The logical operators are also called connectives. DEFINITION 2 Let p and q be propositions. The conjunction of p and q, denoted by pp qq, is the proposition p and q. The conjunction pp qq is true when both p and q are true and is false otherwise. TABLE 2 The Truth Table for the Conjunction of Two Propositions pp qq pp qq T T T T F F F T F F F F EXAMPLE 5 Find the conjunction of the propositions p and q where p is the proposition Today is Friday and q is the proposition It is raining today. Solution: The conjunction of these propositions, pp qq, is the proposition Today is Friday and it is raining today. This proposition is true on rainy Fridays and is false on any day that is not a Friday and on Fridays when it does not rain. DEFINITION 3 Let p and q be propositions. The disjunction of p and q, denoted by pp qq, is the proposition p or q. The disjunction pp qq is false when both p and q are false and is true otherwise. TABLE 3 The Truth Table for the Disjunction of Two Propositions pp qq pp qq T T T T F T F T T F F F 4
5 EXAMPLE 6 Find the disjunction of the propositions p and q where p is the proposition Today is Friday and q is the proposition It is raining today. Solution: The disjunction of these propositions, pp qq, is the proposition Today is Friday or it is raining today. This proposition is true on any day that is either a Friday or a rainy day (including rainy Fridays). It is only false on days that are not Fridays when it also does not rain. DEFINITION 4 Let p and q be propositions. The exclusive or of p and q, denoted by pp qq, is the proposition that is true when exactly one of p and q is true and is false otherwise. Conditional Statements TABLE 4 The Truth Table for the Exclusive OR of Two Propositions pp qq pp qq T T F T F T F T T F F F DEFINITION 5 Let p and q be propositions. The conditional statement, pp qq, is the proposition if p, then q. The conditional statement pp qq is false when p is true and q is false, and true otherwise. In the conditional statement pp qq, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). TABLE 5 The Truth Table for the Conditional Statement of Two Propositions pp qq pp qq T T T T F F F T T F F T The statement pp qq is called a conditional statement because pp qq asserts that q is true on the condition that p holds. A conditional statement is also called an implication. 5
6 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 pp Figure 1. Alternative phrases for pp qq EXAMPLE 7 Let p be the statement Maria learns discrete mathematics and q the statement Maria will find a good job. Express the statement pp qq as a statement in English. Solution: From the definition of conditional statements, we see that when p is the statement Maria learns discrete mathematics and q is the statement Maria will find a good job, pp qq represents the statement If Maria learns discrete mathematics, then she will find a good job. Alternatively: Maria will find a good job when she learns discrete mathematics. For Maria to get a good job, it is sufficient for her to learn discrete mathematics. Maria will find a good job unless she does not learn discrete mathematics. EXAMPLE 8 What is the value of the variable x after the statement if = 4 then xx xx + 1 if xx = 0 before this statement is encountered? (The symbol stands for assignment. The statement xx xx + 1 means the assignment of the value xx + 1 to xx.) Solution: Because = 4 is true, the assignment statement xx xx + 1 is executed. Hence xx has the value = 1 after this statement is encountered. 6
7 CONVERSE, CONTRAPOSITIVE, AND INVERSE Implication pp qq if p, then q Converse of pp qq qq pp if q, then p Contrapositive pp qq qq pp if not q, then not p Inverse pp qq pp qq if not p, then not q Only the contrapositive qq pp, has the same truth values as the implication pp qq. When two compound propositions have the same truth values they are called equivalent. Equivalence is denoted by the operator. EXAMPLE 9 What are the contrapositive, the converse, and the inverse of the conditional statement The home team wins whenever it is raining.? Solution: Because qq whenever pp is one of the ways to express the conditional statement pp qq, the original can be rewritten as If it is raining, then the home team wins. contrapositive qq pp If the home team does not win, then it is not raining. converse qq pp If the home team wins, then it is raining. inverse pp qq If it is not raining, then the home team does not win. Note: Only the contrapositive is equivalent to the original statement. DEFINITION 6 Let p and q be propositions. The biconditional statement, pp qq, is the proposition p if and only if q. The biconditional statement pp qq is true when pp and qq have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications. TABLE 6 The Truth Table for the Biconditional pp qq. pp qq pp qq T T T T F F F T F F F T 7
8 The truth table for pp qq is shown in Table 6. Note that the statement pp qq is true when both the conditional statements pp qq and qq pp are true and is false otherwise. That is why we use the words if and only if to express this logical connective and why it is symbolically written by combining the symbols and. There are other common ways to express pp qq: p is necessary and sufficient for q if p then q, and conversely p iff q. The last way of expressing the biconditional statement pp qq uses the abbreviation iff for if and only if. Note that pp qq has exactly the same truth value as (pp qq) (qq pp). EXAMPLE 10 Let p be the statement You can take the flight and q be the statement You buy a ticket. Express the biconditional pp qq as an English sentence. Solution: You can take the flight if and only if you buy a ticket. Truth Tables of Compound Propositions EXAMPLE 11 Construct the truth table of the compound proposition (pp qq) (pp qq) Solution: 1. Rows: The number of rows in the truth table is equal to 2 vv, where vv is the number of propositional variables. In this case, there are two propositional variables, p and q. Hence, there are 2 2 = 4 rows. 2. Columns. There are as many columns as there are compound propositions, propositional variables, and negated propositional variables. 3. Construction. Complete the columns from left to right moving from the simple to the complex. TABLE 7 The Truth Table of (pp qq) (pp qq) pp qq qq pp qq pp qq (pp qq) (pp qq) T T F T T T T F T T F F F T F F F T F F T T F F 8
9 Precedence of Logical Operators Translating English Sentences TABLE 8 Precedence of Logical Operators Operator Precedence Precedence 1 Highest Lowest EXAMPLE 12 How can this English sentence be translated into a logical expression? You can access the Internet from campus only if you are a computer science major or you are not a freshman. Solution: Define the following propositions: Proposition English statement aa You can access the Internet from campus. cc You are a computer science major. ff You are a freshman only if maps to aa (cc ff) EXAMPLE 13 Translate the following sentence into a logical expression. Logic and Bit Operations You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old. Solution: Define the following propositions: Proposition English statement qq You can ride the roller coaster. rr You are under 4 feet tall. s You are older than 16 years old. (rr ss) qq Truth Value Bit T 1 F 0 9
10 REMARK A bit is the contraction of binary digit, a zero (0) or a one (1). DEFINITION 7 EXAMPLE 12 A bit string is a sequence of zero or more bits. The length of the string is the number of bits in the string is a bit string of length nine. EXAMPLE 13 Find the bitwise OR, bitwise AND, and bitwise XOR, of the bit strings and Solution: OR AND XOR
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