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

Sec. 1.2 Exercise 41Sec. 1.2 Exercise 41• p, q, r 중에 두 개가 참이면 참이 되고 , 그

외의 경우에는 거짓이 되는 복합명제를 구하시오 .

• 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).

Sec. 1.2 Exercise 55Sec. 1.2 Exercise 55• How many different truth tables of compound

propositions are there that involve the propositional variables p and q?

• Solution:16

Sec. 1.3 Exercise 7Sec. 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.

Cont.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.

Sec. 1.3 Exercise 29Sec. 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))

Cont.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.

xy (T(x) T(y) T(x y))

Sec. 1.4 Exercise 1Sec. 1.4 Exercise 1• Translate these statements into English, where

the domain for each variable consists of all real number.

• a) xy(x < y)For every real number x there exists a real number y such that x is less than y.

Cont.Cont.• b) xy(((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) xyz (xy = z)For every real number x and real number y, there exists a real number z such that xy = z.

Sec. 1.4 Exercise 11Sec. 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)

Cont.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)

Cont.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)))

Cont.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)))

Cont.Cont.• h) Some student has never been asked a

question by a faculty member.=> x(S(x) y(F(y) A(y, x)))