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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