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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.)
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.)
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