2. Selected Exercises of Sec. 1.2~Sec.1.4

3. Sec. 1.2 Exercise 41 p, q, r 중에 두 개가 참이면 참이 되고, 그 외 의 경우에는 거짓이 되는 복합명제를 구하 시오. Find a compound proposition involving the propositional variable p, q, and r that is true when exactly two of p, q, and r are true and is false otherwise. Solution: (p  q   r)  (p   q  r)  (  p  q  r).

4. Sec. 1.2 Exercise 55 How many different truth tables of compound propositions are there that involve the propositional variables p and q? Solution:16

5. Sec. 1.3 Exercise 7 Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people. a)  x (C(x)  F(x)) Every comedian is funny. b)  x (C(x)  F(x)) Every person is a funny comedian.

6. Cont. c)  x (C(x)  F(x)) There exists a person such that if she or he is a comedian, then she or he is funny. d)  x (C(x)  F(x)) Some comedians are funny.

7. Sec. 1.3 Exercise 29 Express each of these statements using logical operators, predicates, and quantifiers. Solution: Let T(x) mean that x is a tautology and C(x) mean that x is a contradiction. a) Some propositions are tautologies. :  x T(x) b) The negation of a contradiction is a tautology  x (C(x)  T(  x))

8. Cont. c) The disjunction of two contingencies can be a tautology.  x  y(  T(x)   C(x)   T(y)   C(y)  T(x  y)) c) The conjunction of two tautologies is a tautology.  x  y (T(x)  T(y)  T(x  y))

9. Sec. 1.4 Exercise 1 Translate these statements into English, where the domain for each variable consists of all real number. a)  x  y(x < y) For every real number x there exists a real number y such that x is less than y.

10. Cont. b)  x  y(((x ≥ 0)  (y ≥ 0))  xy ≥ 0)) For every real number x and real number y, if x and y are both nonnegative, then their product is nonnegative. c)  x  y  z (xy = z) For every real number x and real number y, there exists a real number z such that xy = z.

11. Sec. 1.4 Exercise 11 Let S(x) be the predicate “x is a student,” F(x) the predicate “x is faculty member,” and A(x, y) the predicate “x has asked y a question,” where the domain consists of all people associated with your school. Use quantifiers to express each of these statements. a) Lois has asked Professor Michaels a question. => A(Lois, Professor Michaels)

12. Cont. b) Every student has asked Professor Gross a question. =>  x(S(x)  A(x, Professor Gross) c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller. =>  x(F(x)  (A(x, Professor Miller)  A(Professor Miller, x)

13. Cont. d) Some student has not asked any faculty member a question. =>  x(S(x)   y(F(y)   A(x, y))) e) There is a faculty member who has never been asked a question by a student. =>  x(F(x)   y(S(y)   A(y, x)))

14. Cont. f) Some student has asked every faculty member a question. =>  y(F(y)   x(S(x)  A(y, x))) g) There is a faculty member who has asked every other faculty member a question. =>  x(F(x)   y((F(y)  (y≠ x))  A(x, y)))

15. Cont. h) Some student has never been asked a question by a faculty member. =>  x(S(x)   y(F(y)   A(y, x)))