w that the following properties hold for all elements x , y , and z of a lattice L. a) x ∧ y = y ∧ x and x ∨ y = y ∨ x (commutative laws) b) ( x ∧ y ) ∧ z = x ∧ ( y ∧ z ) and ( x ∨ y ) ∨ z = x ∨ ( y ∨ z ) (associative laws) c) x ∧ ( x ∨ y ) = x and x ∨ ( x ∧ y ) = x (absorption laws) d) x ∧ x = x and x ∨ x = x (idempotent laws)
w that the following properties hold for all elements x , y , and z of a lattice L. a) x ∧ y = y ∧ x and x ∨ y = y ∨ x (commutative laws) b) ( x ∧ y ) ∧ z = x ∧ ( y ∧ z ) and ( x ∨ y ) ∨ z = x ∨ ( y ∨ z ) (associative laws) c) x ∧ ( x ∨ y ) = x and x ∨ ( x ∧ y ) = x (absorption laws) d) x ∧ x = x and x ∨ x = x (idempotent laws)
3.
Let S(x) be the statement “x is a student," P(x) be the statement “x is a
professor," and Q(x, y) be the statement “x asked a question to y." If the domain of x and y
consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y),
quantifiers, and logical operators.
Example: Some student has asked every professor a question.
Answer.
3x(S(x) A Vy(P(V) → Q(x,y)))
a. Every professor has been asked a question by some student.
b. There is a student who has never asked a question to a professor.
c. There is a professor who has been asked a question by exactly one student.
d. There are two different students who have asked each other a question.
3.
Let S(x) be the statement “x is a student," P(x) be the statement “x is a
professor," and Q(x, y) be the statement “x asked a question to y." If the domain of x and y
consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y),
quantifiers, and logical operators.
Example: Some student has asked every professor a question.
Answer:
ax(S(x) A Vy(P(y)→ Q(x, y)))
a. Every professor has been asked a question by some student.
b. There is a professor who has never been asked a question by any student.
c. There is a student who has asked a question to exactly one professor.
d. There are two different students who have asked each other a question.
Chapter 9 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
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