How does x differ in what it represents in the following statements? x is real. x represents no values

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1 Section. redicate Logic Discussion: In Maths we use variables usuall ranging over numbers in various was. How does differ in what it represents in the following statements? is real. = 0 represents one value = 0 > represents some but not all values + 0 = represents all values + = 0 represents no values Definition: redicate predicate is a sentence that contains one or more variables and becomes a statement when specific values are substituted for the variables. Definition: Domain The domain of a predicate variable consists of all values that ma be substituted in place of the variable WUCT Logic 60

2 Definition: Truth Set If is a predicate and has domain D the truth set of is the set of all elements of D that make true. The truth set is denoted { D : } and is read the set of all in D such that. Eamples: Let be the predicate > domain the set of real numbers. with i.e. Write down and indicate which are true and which are false. Determine the truth set of : : : > > > or or or 4 > > 4 > True False True { : > } = { : < 0 > } Let Qn be the predicate n is factor of 8. Determine the truth set of Qn if 8 = ± ± 8 { n + 8 = ± ± 4 + n :" n isa factor of 8"} = {48} WUCT Logic 6

3 Eercises: Let be the predicate > domain the set of integers. with i.e. Write down 0 and indicate which are true and which are false. Determine the truth set of Let Qn be the predicate n is factor of 6. Determine the truth set of Qn if n WUCT Logic 6

4 .. Quantifiers wa to obtain statements from predicates is to add quantifiers. Quantifiers are words that refer to quantities such as all ever or some and tell for how man elements a given predicate is true.... Universal Quantifier The smbol denotes for all and is called the universal quantifier. Definition: Universal Statement Let be a predicate and D the domain of. universal statement is a statement of the form D. It is defined to be true if and onl if is true for ever in D. It is defined to be false if and onl if is false for at least one in D. value of for which is false is called a countereample to the universal statement. Eamples: Write the sentence ll human beings are mortal using the universal quantifier. Let H be the set of human beings. h Hh is mortal WUCT Logic 6

5 Consider = { }. With the following must hold: Thus there will be predicates which must hold. Eercises: Write the following statements using the universal quantifier. Determine whether each statement is true or false. ll dogs are animals The square of an real number is positive. Ever integer is a rational number. WUCT Logic 64

6 Eercises: Write the following statements in words. Determine whether each statement is true or false.. WUCT Logic 65

7 ... Eistential Quantifier The smbol denotes there eists and is called the eistential quantifier. Definition: Eistential Statement Let be a predicate and D the domain of. n eistential statement is a statement of the form D. It is defined to be true if and onl if is true for at least one in D. It is defined to be false if and onl if is false for all in D. Eamples: Write the sentence Some people are vegetarians using the eistential quantifier. Let H be the set of human beings. h Hh is a vegetarian Consider = { }. With the following must hold: Thus there will be predicate which must hold. WUCT Logic 66

8 Eercises: Write the following statements using the eistential quantifier. Determine whether each statement is true or false. Some cats are black There is a real number whose square is negative. Some programs are structured. WUCT Logic 67

9 Eercises: Write the following statements in words. Determine whether each statement is true or false. m m = m =. WUCT Logic 68

10 ... Negation of Universal Statements Let be a predicate and D the domain of. The negation of a universal statement of the form: D is logicall equivalent to D ~ Smbolicall ~ D D~ Eample: Write down the negation of the following statement. + Negation: False. ~ ~ < WUCT Logic 69

11 Eercises: Write down the negation of the following statement. 0 Write down the negation of the following statement. + 0 < WUCT Logic 70

12 Eample: Write the following statement using quantifiers. Find its negation and determine whether the statement or its negation is true giving a brief reason.. Ever real number is either positive or negative. Statement: < 0 > 0 Negation: ~ < 0 > 0 ~ ~ = 0 < 0 > 0 < 0 0 ~ > 0 0 The true statement is the negation because = 0 is neither positive nor negative. WUCT Logic 7

13 Eercises: Write the following statement using quantifiers. Find the negation. The square of an integer is positive. Write the following statement using quantifiers. Find the negation. ll computer programs are finite. WUCT Logic 7

14 ..4. Negation of Eistential Quantifiers Let be a predicate and D the domain of. The negation of an eistential statement of the form: D is logicall equivalent to D ~ Smbolicall ~ D D~ Eample: Write down the negation of the following statement. = Negation: ~ = ~ = The negation is true. WUCT Logic 7

15 Eercises: Write down the negation of the following statement. z z is odd z is even Write down the negation of the following statement. n n is even n is prime WUCT Logic 74

16 Eample: Write the following statement using quantifiers. Find its negation Some dogs are vegetarians. Let D be the set of dogs. Statement: d D d is vegetarian Negation: ~ d D d is vegetarian d D ~ d is vegetarian d D d is not vegetarian ll dogs are not vegetarian Eercises: Write the following statement using quantifiers. Find the negation. There is a real number that is rational. WUCT Logic 75

17 Write the following statement using quantifiers. Find the negation. Some computer hackers are over 40. Write the following statement using quantifiers. Find the negation. Some animals are dogs. WUCT Logic 76

18 ..5. Multiple Quantifiers When a statement contains multiple quantifiers their order must be applied as written and will produce different results for the truth set. Eamples: Write the following statements using quantifiers: Everbod loves somebod. Let H be the set of people. Statement: H H loves. Somebod loves everone. Let H be the set of people. Statement: H H loves. WUCT Logic 77

19 Eercises: Write the following statements using quantifiers: Everbod loves everbod. The Commutative Law of ddition for Everone had a mother. There is an oldest person. WUCT Logic 78

20 Eamples: Write the following statements without using quantifiers: + = 0 Statement: Given an real number ou can find a real number so that the sum of the two is zero. lternativel: Ever real number has an additive inverse. + = Statement: There is a real number which added to an other real number results in the other number. lternativel: Ever real number has an additive identit. Eercises: Write the following statements without using quantifiers: c colours a animals a is coloured c b books p people p has read b WUCT Logic 79

21 WUCT Logic Interpreting Statements with Multiple Quantifiers To establish the truth of a statement with more than one quantifier take the action suggested b the quantifiers as being performed in the order in which the quantifiers occur. Consider } { } { = = and the predicate. There will be 6 possible predicates:. For to be true the following must hold: Thus there will be 6 predicates which must all be true. That is for all pairs must be true. It will be false if there is one pair for which is false.

22 WUCT Logic 8 For to be true the following must hold: Thus there will be predicates which must be true. That is for ever there must be at least one so that is true. Given an element in ou can find an element in so that is true. It will be false if there is one in for which is false for ever in. For to be true the following must hold: Thus there will be predicates which must be true. That is there is one that when paired with an is true. You can find one element in that with all elements in is true. It will be false if for ever in there is a in for which is false.

23 For to be true the following must hold: Thus there will be predicate which must be true. That is there is one that when paired with one is true. You can find one element in and one element in is true. It will be false if for all pairs is false. Summar: Statement When true? When false? is true for all pairs There is a pair for which is false For ever there is a for which is true There is an such that is false for ever There is an such that is true for ever For ever there is a for which is false There is a pair for which is false for all pairs is true WUCT Logic 8

24 WUCT Logic Negation of Statements with Multiple Quantifiers. To negate statements with multiple quantifiers each quantifier is negated and the predicate must be negated. To negate ~ ~ To negate ~ ~ To negate ~ ~ To negate ~ ~ Eamples: Write the negation of the following: Statement: 0 = + Negation: 0 then Take False: 0 0 ~ = = + = + = +

25 Statement: = Negation: ~ = True : Take = then = Eercises: Write the negation of the following: Statement: c colours a animals a is coloured c Statement: b books p people p has read b WUCT Logic 84