On this educational webpage, we present a logical proof exercise that involves operations with quantifiers and logical connectives. The problem to be solved is as follows: ### 2. Prove: ##### a. \[ \forall x \ q(x) \equiv \neg [\exists x \ \neg q(x)] \] ##### b. \[ \exists x \ (\neg p(x) \land \neg q(x)) \equiv \neg [\forall x \ (p(x) \lor q(x))] \] ### Explanation: In these logical statements, the goal is to prove the equivalence of two expressions. #### Part (a): - The statement reads: "For all \( x \), \( q(x) \) if and only if it is not the case that there exists some \( x \) for which \( q(x) \) does not hold." #### Part (b): - The statement reads: "There exists an \( x \) such that both \( p(x) \) does not hold and \( q(x) \) does not hold if and only if it is not the case that for all \( x \), either \( p(x) \) or \( q(x) \) holds." These statements involve understanding the relationship between universal quantification ( \( \forall \) ), existential quantification ( \( \exists \) ), negation ( \( \neg \) ), conjunction ( \( \land \) ), and disjunction ( \( \lor \) ). This exercise helps to practice and comprehend the logical manipulation of quantifiers and logical connectives.
On this educational webpage, we present a logical proof exercise that involves operations with quantifiers and logical connectives. The problem to be solved is as follows: ### 2. Prove: ##### a. \[ \forall x \ q(x) \equiv \neg [\exists x \ \neg q(x)] \] ##### b. \[ \exists x \ (\neg p(x) \land \neg q(x)) \equiv \neg [\forall x \ (p(x) \lor q(x))] \] ### Explanation: In these logical statements, the goal is to prove the equivalence of two expressions. #### Part (a): - The statement reads: "For all \( x \), \( q(x) \) if and only if it is not the case that there exists some \( x \) for which \( q(x) \) does not hold." #### Part (b): - The statement reads: "There exists an \( x \) such that both \( p(x) \) does not hold and \( q(x) \) does not hold if and only if it is not the case that for all \( x \), either \( p(x) \) or \( q(x) \) holds." These statements involve understanding the relationship between universal quantification ( \( \forall \) ), existential quantification ( \( \exists \) ), negation ( \( \neg \) ), conjunction ( \( \land \) ), and disjunction ( \( \lor \) ). This exercise helps to practice and comprehend the logical manipulation of quantifiers and logical connectives.
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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![On this educational webpage, we present a logical proof exercise that involves operations with quantifiers and logical connectives. The problem to be solved is as follows:
### 2. Prove:
##### a.
\[ \forall x \ q(x) \equiv \neg [\exists x \ \neg q(x)] \]
##### b.
\[ \exists x \ (\neg p(x) \land \neg q(x)) \equiv \neg [\forall x \ (p(x) \lor q(x))] \]
### Explanation:
In these logical statements, the goal is to prove the equivalence of two expressions.
#### Part (a):
- The statement reads: "For all \( x \), \( q(x) \) if and only if it is not the case that there exists some \( x \) for which \( q(x) \) does not hold."
#### Part (b):
- The statement reads: "There exists an \( x \) such that both \( p(x) \) does not hold and \( q(x) \) does not hold if and only if it is not the case that for all \( x \), either \( p(x) \) or \( q(x) \) holds."
These statements involve understanding the relationship between universal quantification ( \( \forall \) ), existential quantification ( \( \exists \) ), negation ( \( \neg \) ), conjunction ( \( \land \) ), and disjunction ( \( \lor \) ). This exercise helps to practice and comprehend the logical manipulation of quantifiers and logical connectives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bc7733f-239b-44d3-8932-ec574e8191d8%2F8920eb30-8336-4ec2-bd3d-50294db3e44d%2F6mx96ay_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this educational webpage, we present a logical proof exercise that involves operations with quantifiers and logical connectives. The problem to be solved is as follows:
### 2. Prove:
##### a.
\[ \forall x \ q(x) \equiv \neg [\exists x \ \neg q(x)] \]
##### b.
\[ \exists x \ (\neg p(x) \land \neg q(x)) \equiv \neg [\forall x \ (p(x) \lor q(x))] \]
### Explanation:
In these logical statements, the goal is to prove the equivalence of two expressions.
#### Part (a):
- The statement reads: "For all \( x \), \( q(x) \) if and only if it is not the case that there exists some \( x \) for which \( q(x) \) does not hold."
#### Part (b):
- The statement reads: "There exists an \( x \) such that both \( p(x) \) does not hold and \( q(x) \) does not hold if and only if it is not the case that for all \( x \), either \( p(x) \) or \( q(x) \) holds."
These statements involve understanding the relationship between universal quantification ( \( \forall \) ), existential quantification ( \( \exists \) ), negation ( \( \neg \) ), conjunction ( \( \land \) ), and disjunction ( \( \lor \) ). This exercise helps to practice and comprehend the logical manipulation of quantifiers and logical connectives.
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