Examples. Example (1) Example (2) Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Solve. x 2 + (y 1) 2 = 0.

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1 Examples Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Example (1) Solve x 2 + (y 1) 2 = 0. The solution is x = 0 AND y = 1. [p q.] Example (2) Solve x(y 1) = 0.

2 Examples Let x, y be two variables, and denote statements p : x = 0 and q : y = 1. Example (1) Solve x 2 + (y 1) 2 = 0. The solution is x = 0 AND y = 1. [p q.] Example (2) Solve x(y 1) = 0. The solution is x = 0 OR y = 1. [p q.]

3 Negation Given a statement p, we call not p the negation of p, and denote it by p. It is clear ( p) = p, as two statements.

4 Negation Given a statement p, we call not p the negation of p, and denote it by p. It is clear ( p) = p, as two statements. Do negate the whole statement, instead of a single word of it: the negation of It sometimes rains. is It never rains., not It sometimes doesn t rain.!!!

5 Negation Given a statement p, we call not p the negation of p, and denote it by p. It is clear ( p) = p, as two statements. Do negate the whole statement, instead of a single word of it: the negation of It sometimes rains. is It never rains., not It sometimes doesn t rain.!!! The truth value of p is always opposite to that of p.

6 Negation Given a statement p, we call not p the negation of p, and denote it by p. It is clear ( p) = p, as two statements. Do negate the whole statement, instead of a single word of it: the negation of It sometimes rains. is It never rains., not It sometimes doesn t rain.!!! The truth value of p is always opposite to that of p. When used after a conjunction or disjunction, there are De Morgan s Laws: (p q) = ( p) ( q); (p q) = ( p) ( q).

7 Examples Continued Example (3) Solve x 2 + (y 1) 2 0.

8 Examples Continued Example (3) Solve x 2 + (y 1) 2 0. The solution is x, y being any real numbers, EXCLUDING x = 0 AND y = 1. [ (p q).] In other words, it is x 0 OR y 1. [( p) ( q).]

9 Examples Continued Example (3) Solve x 2 + (y 1) 2 0. The solution is x, y being any real numbers, EXCLUDING x = 0 AND y = 1. [ (p q).] In other words, it is x 0 OR y 1. [( p) ( q).] Example (4) Solve x(y 1) 0.

10 Examples Continued Example (3) Solve x 2 + (y 1) 2 0. The solution is x, y being any real numbers, EXCLUDING x = 0 AND y = 1. [ (p q).] In other words, it is x 0 OR y 1. [( p) ( q).] Example (4) Solve x(y 1) 0. The solution is EVERYTHING EXCEPT x = 0 OR y = 1. [ (p q).] Namely, it is x 0 AND y 1. [( p) ( q).]

11 Exercise: Translating Symbols into Words Let p and q be statements as follows: p: Aggies beat LSU Tigers last year. q: Bush 41 was laid to rest at his presidential library.

12 Exercise: Translating Symbols into Words Let p and q be statements as follows: p: Aggies beat LSU Tigers last year. q: Bush 41 was laid to rest at his presidential library. Question 1: what does ( p) q mean?

13 Exercise: Translating Symbols into Words Let p and q be statements as follows: p: Aggies beat LSU Tigers last year. q: Bush 41 was laid to rest at his presidential library. Question 1: what does ( p) q mean? Answer: Aggies didn t beat LSU Tigers last year, or Bush 41 was laid to rest at his presidential library.

14 Exercise: Translating Symbols into Words Let p and q be statements as follows: p: Aggies beat LSU Tigers last year. q: Bush 41 was laid to rest at his presidential library. Question 1: what does ( p) q mean? Answer: Aggies didn t beat LSU Tigers last year, or Bush 41 was laid to rest at his presidential library. Question 2: what does p ( q) mean?

15 Exercise: Translating Symbols into Words Let p and q be statements as follows: p: Aggies beat LSU Tigers last year. q: Bush 41 was laid to rest at his presidential library. Question 1: what does ( p) q mean? Answer: Aggies didn t beat LSU Tigers last year, or Bush 41 was laid to rest at his presidential library. Question 2: what does p ( q) mean? Answer: Aggies beat LSU Tigers last year, but (= and) Bush 41 was not laid to rest at his presidential library.

16 Exercise: Translating Words into Symbols Question 1: Symbolize statement An Aggie does not cheat, or tolerate whose who do. with proper choices of p, q.

17 Exercise: Translating Words into Symbols Question 1: Symbolize statement An Aggie does not cheat, or tolerate whose who do. with proper choices of p, q. Answer: (p q) or ( p) ( q), with p: An Aggie cheats., and q: An Aggie tolerates cheaters.

18 Exercise: Translating Words into Symbols Question 1: Symbolize statement An Aggie does not cheat, or tolerate whose who do. with proper choices of p, q. Answer: (p q) or ( p) ( q), with p: An Aggie cheats., and q: An Aggie tolerates cheaters. Question 2: Symbolize statement A calculator is allowed in exams, only if it is incapable to perform symbolic mathematics, and the user has asked me for permission. with p: Calculators with symbolic functions are allowed in exams. and q: Calculators need special permission to be used in exams.

19 Exercise: Translating Words into Symbols Question 1: Symbolize statement An Aggie does not cheat, or tolerate whose who do. with proper choices of p, q. Answer: (p q) or ( p) ( q), with p: An Aggie cheats., and q: An Aggie tolerates cheaters. Question 2: Symbolize statement A calculator is allowed in exams, only if it is incapable to perform symbolic mathematics, and the user has asked me for permission. with p: Calculators with symbolic functions are allowed in exams. and q: Calculators need special permission to be used in exams. Answer: ( p) q.

20 Summary For this section, you are supposed to know how to: Determine if a sentence is a statement;

21 Summary For this section, you are supposed to know how to: Determine if a sentence is a statement; Judge the truth value of a compound statement based on that of its generators;

22 Summary For this section, you are supposed to know how to: Determine if a sentence is a statement; Judge the truth value of a compound statement based on that of its generators; Translate between two languages.

23 L2: Truth Tables Jiakun Pan Jan 16, 2019

24 Motivation Let p and q be stated as follows: p: Blah-blah-blah ; q: Abracadabra. Q: How many statements below are true? p q, p q, p, and q.

25 Motivation Let p and q be stated as follows: p: Blah-blah-blah ; q: Abracadabra. Q: How many statements below are true? p q, p q, p, and q. A:2, always.

26 Motivation Let p and q be stated as follows: p: Blah-blah-blah ; q: Abracadabra. Q: How many statements below are true? p q, p q, p, and q. A:2, always. Such things happen frequently in logic, and truth tables are tools that help you summarize.

27 A First Example Let us begin with the truth table for p ( p).

28 A First Example Let us begin with the truth table for p ( p). Given blank table p p p ( p) T F,

29 A First Example Let us begin with the truth table for p ( p). Given blank table p p p ( p) T F, we can firstly determine the truth value of p, obtaining

30 A First Example Let us begin with the truth table for p ( p). Given blank table p p p ( p) T F, we can firstly determine the truth value of p, obtaining p p p ( p) T F, F T

31 A First Example Let us begin with the truth table for p ( p). Given blank table p p p ( p) T F, we can firstly determine the truth value of p, obtaining p p p ( p) T F F T, and finally the conjunction p p p ( p) T F F F T F.

32 A First Example (Continued) So if you know this table well, then you don t have to judge p and p each time, if you only want to know whether p ( p) is false.

33 A First Example (Continued) So if you know this table well, then you don t have to judge p and p each time, if you only want to know whether p ( p) is false. If a statement is unconditionally false, then we call the statement a contradiction. By definition, p ( p) is one.

34 A First Example (Continued) So if you know this table well, then you don t have to judge p and p each time, if you only want to know whether p ( p) is false. If a statement is unconditionally false, then we call the statement a contradiction. By definition, p ( p) is one. Similarly, you can draw the truth table for p ( p), and see that it is always true.

35 A First Example (Continued) So if you know this table well, then you don t have to judge p and p each time, if you only want to know whether p ( p) is false. If a statement is unconditionally false, then we call the statement a contradiction. By definition, p ( p) is one. Similarly, you can draw the truth table for p ( p), and see that it is always true. If a statement is regardlessly true, then we call it a tautology. As you can see, p ( p) fits the definition well.