2. Recall that, by definition, {A1, A2,..., An} = A if and only if every model of {A1, A2,..., An} is a model of A. (a) What does it mean for an interpretation to be a model of {A1, A2,..., An}? (b) In class, we said that {A₁, A2,..., An} = A if and only if A₁ A₂ A. ^ An → A is valid. In parts (b) and (c) of this question, you will justify this statement. First, suppose that every model of {A₁, A2,..., An} is a model for A. Under this as- sumption, show that A₁ A2 AA An→ A is valid. (Hint: Let I be an arbitrary interpretation and show that vy(A1AA2AA An → A) = T.) (c) Now suppose that A₁ A A2 AA An→ A is valid. Under this assumption, show that any model of {A₁, A2,..., An} is a model of A. (Hint: Let be an arbitrary model of {A1, A2,..., An} and show that must also be a model of A.)
2. Recall that, by definition, {A1, A2,..., An} = A if and only if every model of {A1, A2,..., An} is a model of A. (a) What does it mean for an interpretation to be a model of {A1, A2,..., An}? (b) In class, we said that {A₁, A2,..., An} = A if and only if A₁ A₂ A. ^ An → A is valid. In parts (b) and (c) of this question, you will justify this statement. First, suppose that every model of {A₁, A2,..., An} is a model for A. Under this as- sumption, show that A₁ A2 AA An→ A is valid. (Hint: Let I be an arbitrary interpretation and show that vy(A1AA2AA An → A) = T.) (c) Now suppose that A₁ A A2 AA An→ A is valid. Under this assumption, show that any model of {A₁, A2,..., An} is a model of A. (Hint: Let be an arbitrary model of {A1, A2,..., An} and show that must also be a model of A.)
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
Transcribed Image Text:2. Recall that, by definition, {A1, A2,..., An} = A if and only if every model of {A1, A2,..., An}
is a model of A.
(a) What does it mean for an interpretation
to be a model of {A1, A2,..., An}?
(b) In class, we said that {A₁, A2,..., An} = A if and only if A₁ A₂ ^ ^ An → A is valid.
In parts (b) and (c) of this question, you will justify this statement.
First, suppose that every model of {A₁, A2,..., An} is a model for A. Under this as-
sumption, show that A₁ A2 AA An→ A is valid. (Hint: Let I be an arbitrary
interpretation and show that u(A₁AA2AA An → A) = T.)
(c) Now suppose that A₁ A A2 AA An→ A is valid. Under this assumption, show that
any model of {A₁, A2,..., An} is a model of A. (Hint: Let be an arbitrary model of
{A1, A2,..., An} and show that
must also be a model of A.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
