Justify (argue, prove, ...) that the following formula is valid Vx (A(x)→B(x)) → (Vx A(x)→V¢B(x))
Justify (argue, prove, ...) that the following formula is valid Vx (A(x)→B(x)) → (Vx A(x)→V¢B(x))
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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![**Explanation of Formula Validity**
We are tasked with proving the validity of the following logical formula:
\[
\forall x \ (A(x) \rightarrow B(x)) \rightarrow (\forall x \ A(x) \rightarrow \forall x \ B(x))
\]
**Explanation:**
1. **Understanding the Formula:**
- The expression consists of an implication (indicated by \(\rightarrow\)) between two universally quantified statements (indicated by \(\forall x\)).
- The left part of the implication: \(\forall x \ (A(x) \rightarrow B(x))\), asserts that for every element \(x\), if \(A(x)\) is true, then \(B(x)\) is true.
- The right part of the implication: \(\forall x \ A(x) \rightarrow \forall x \ B(x)\), suggests that if \(A(x)\) is true for all \(x\), then \(B(x)\) must also be true for all \(x\).
2. **Proof Outline:**
- Assume the premise \(\forall x \ (A(x) \rightarrow B(x))\) is true.
- We need to show that from this assumption, \(\forall x \ A(x) \rightarrow \forall x \ B(x)\) follows.
- Assume \(\forall x \ A(x)\) (for each \(x\), \(A(x)\) is true). Given this assumption and the premise, it follows that \(B(x)\) must also be true for each such \(x\) as \(A(x) \rightarrow B(x)\).
- Therefore, \(\forall x \ B(x)\) holds.
In conclusion, the formula is valid as the premise logically ensures the conclusion through universal instantiation and modus ponens.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74d35d78-10f2-4c7e-8a1f-3d0300cedc60%2Ff12b5c18-b3ac-4540-a529-91aecacee5d3%2Fchya1qs_processed.png&w=3840&q=75)
Transcribed Image Text:**Explanation of Formula Validity**
We are tasked with proving the validity of the following logical formula:
\[
\forall x \ (A(x) \rightarrow B(x)) \rightarrow (\forall x \ A(x) \rightarrow \forall x \ B(x))
\]
**Explanation:**
1. **Understanding the Formula:**
- The expression consists of an implication (indicated by \(\rightarrow\)) between two universally quantified statements (indicated by \(\forall x\)).
- The left part of the implication: \(\forall x \ (A(x) \rightarrow B(x))\), asserts that for every element \(x\), if \(A(x)\) is true, then \(B(x)\) is true.
- The right part of the implication: \(\forall x \ A(x) \rightarrow \forall x \ B(x)\), suggests that if \(A(x)\) is true for all \(x\), then \(B(x)\) must also be true for all \(x\).
2. **Proof Outline:**
- Assume the premise \(\forall x \ (A(x) \rightarrow B(x))\) is true.
- We need to show that from this assumption, \(\forall x \ A(x) \rightarrow \forall x \ B(x)\) follows.
- Assume \(\forall x \ A(x)\) (for each \(x\), \(A(x)\) is true). Given this assumption and the premise, it follows that \(B(x)\) must also be true for each such \(x\) as \(A(x) \rightarrow B(x)\).
- Therefore, \(\forall x \ B(x)\) holds.
In conclusion, the formula is valid as the premise logically ensures the conclusion through universal instantiation and modus ponens.
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