a. The wff whenever the antecedent is true, so is the consequent, and the implication is there- fore true. (Vx)P(x) → Ax)P(x) b. The wff is valid. In any interpretation, if every element of the domain has a certain prop- erty, then there exists an element of the domain that has that property. (Remember that the domain of any interpretation must have at least one object in it.) Therefore, (Vx)P(x) → P(a) is valid because in any interpretation, a is a particular member of the domain and therefore has the property that is shared by all members of the domain. c. The wff (Vx)[P(x) A Q(x)]→(Vx)P(x) ^ (Vx)Q(x) is valid. If both P and Q are true for all the elements of the domain, then P is true for all elements and Q is true for all elements, and vice versa. d. The wff P(x) → [Q(x) → P(x)] is valid, even though it contains a free variable. To see this, consider any inter- pretation, and let x be any member of the domain. Then x either does or does not have property P. If x does not have property P, then P(x) is false; because P(x) is the antecedent of the main inmplication, this implication is true. If x does have property P, then P(x) is true; regardless of the truth value of Q(x), the implication Q (x) → P(x) is true, and so the main implication is also true. e. The wff (Ex)P(x) → (Vx)P(x) is not valid. For example, in the interpretation where the domain consists of the integers and P(x) means that x is even, it is true that there exists an integer that is even, but it is false that every integer is even. The antecedent of the implication is true and the consequent is false, so the value of the implication is false.
a. The wff whenever the antecedent is true, so is the consequent, and the implication is there- fore true. (Vx)P(x) → Ax)P(x) b. The wff is valid. In any interpretation, if every element of the domain has a certain prop- erty, then there exists an element of the domain that has that property. (Remember that the domain of any interpretation must have at least one object in it.) Therefore, (Vx)P(x) → P(a) is valid because in any interpretation, a is a particular member of the domain and therefore has the property that is shared by all members of the domain. c. The wff (Vx)[P(x) A Q(x)]→(Vx)P(x) ^ (Vx)Q(x) is valid. If both P and Q are true for all the elements of the domain, then P is true for all elements and Q is true for all elements, and vice versa. d. The wff P(x) → [Q(x) → P(x)] is valid, even though it contains a free variable. To see this, consider any inter- pretation, and let x be any member of the domain. Then x either does or does not have property P. If x does not have property P, then P(x) is false; because P(x) is the antecedent of the main inmplication, this implication is true. If x does have property P, then P(x) is true; regardless of the truth value of Q(x), the implication Q (x) → P(x) is true, and so the main implication is also true. e. The wff (Ex)P(x) → (Vx)P(x) is not valid. For example, in the interpretation where the domain consists of the integers and P(x) means that x is even, it is true that there exists an integer that is even, but it is false that every integer is even. The antecedent of the implication is true and the consequent is false, so the value of the implication is false.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![a. The wff
whenever the antecedent is true, so is the consequent, and the implication is there-
fore true.
(Vx)P(x) → Ax)P(x)
b. The wff
is valid. In any interpretation, if every element of the domain has a certain prop-
erty, then there exists an element of the domain that has that property. (Remember
that the domain of any interpretation must have at least one object in it.) Therefore,
(Vx)P(x) → P(a)
is valid because in any interpretation, a is a particular member of the domain and
therefore has the property that is shared by all members of the domain.
c. The wff
(Vx)[P(x) A Q(x)]→(Vx)P(x) ^ (Vx)Q(x)
is valid. If both P and Q are true for all the elements of the domain, then P is true
for all elements and Q is true for all elements, and vice versa.
d. The wff
P(x) → [Q(x) → P(x)]
is valid, even though it contains a free variable. To see this, consider any inter-
pretation, and let x be any member of the domain. Then x either does or does not
have property P. If x does not have property P, then P(x) is false; because P(x)
is the antecedent of the main inmplication, this implication is true. If x does have
property P, then P(x) is true; regardless of the truth value of Q(x), the implication
Q (x) → P(x) is true, and so the main implication is also true.
e. The wff
(Ex)P(x) → (Vx)P(x)
is not valid. For example, in the interpretation where the domain consists of the
integers and P(x) means that x is even, it is true that there exists an integer that is
even, but it is false that every integer is even. The antecedent of the implication is
true and the consequent is false, so the value of the implication is false.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd588b409-f5e1-425b-8518-96db1abda83e%2Fab4c0440-c3ef-4c9c-ad8e-02380a2e3a64%2F42lnn48.png&w=3840&q=75)
Transcribed Image Text:a. The wff
whenever the antecedent is true, so is the consequent, and the implication is there-
fore true.
(Vx)P(x) → Ax)P(x)
b. The wff
is valid. In any interpretation, if every element of the domain has a certain prop-
erty, then there exists an element of the domain that has that property. (Remember
that the domain of any interpretation must have at least one object in it.) Therefore,
(Vx)P(x) → P(a)
is valid because in any interpretation, a is a particular member of the domain and
therefore has the property that is shared by all members of the domain.
c. The wff
(Vx)[P(x) A Q(x)]→(Vx)P(x) ^ (Vx)Q(x)
is valid. If both P and Q are true for all the elements of the domain, then P is true
for all elements and Q is true for all elements, and vice versa.
d. The wff
P(x) → [Q(x) → P(x)]
is valid, even though it contains a free variable. To see this, consider any inter-
pretation, and let x be any member of the domain. Then x either does or does not
have property P. If x does not have property P, then P(x) is false; because P(x)
is the antecedent of the main inmplication, this implication is true. If x does have
property P, then P(x) is true; regardless of the truth value of Q(x), the implication
Q (x) → P(x) is true, and so the main implication is also true.
e. The wff
(Ex)P(x) → (Vx)P(x)
is not valid. For example, in the interpretation where the domain consists of the
integers and P(x) means that x is even, it is true that there exists an integer that is
even, but it is false that every integer is even. The antecedent of the implication is
true and the consequent is false, so the value of the implication is false.
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