In class, I asserted that is the same as , where and are predicates. That is, you can "not" or "negate" a predicate that is itself a conjunction of predicates by distributing the "not" over the inner predicates, and changing the "and" to an "or".

Using this rule, and the fact that is equivalent to and also that is equivalent to , we can argue that can be re-written as what?

Question

In class, I asserted that  is the same as , where  and  are predicates. That is, you can "not" or "negate" a predicate that is itself a conjunction of predicates by distributing the "not" over the inner predicates, and changing the "and" to an "or".

Using this rule, and the fact that  is equivalent to  and also that  is equivalent to , we can argue that  can be re-written as what?

BlackTom AI · Accepted Answer

The prompt asks us to rewrite the negation of a conjunction of predicates using the given logical equivalences, but the provided data has no answer options to analyze. I will nevertheless walk through the correct transformation and point out common missteps that might appear.

First, recall De Morgan’s law for predicates: not (A and B) is equivalent to (not A) or (not B). When the domain involves a quantified variable x, and A and B are predicates depending on x (e.g., p(x) and q(x)), this distributes inside the negation but also interacts with the quantifier.

If we start with not exists x of (p(x) and q(x)), i.e., ¬∃x [p(x) ∧ q(x)], we can move the negation inward across the existential quantifier: ¬∃x [p(x) ∧ q(x)] ≡ ∀x ¬[p(x) ∧ q(x)]. This step uses the standard equivalence ¬∃x φ ≡ ∀x ¬φ.

Then apply De Morgan’s law to ¬[p(x) ∧ q(x)]: ¬[p(x) ∧ q(x)] ≡ (¬p(x)) ∨ (¬q(x)).

Putting these together gives the full rewrite: ¬∃x [p(x) ∧ q(x)] ≡ ∀x [¬p(x) ∨ ¬q(x)].

Now, evaluating the proposed candidate answer (if we treat it as the option given in the data): ¬ ∃ x ( ¬ p(x) ∨ ¬ q(x) ). This would read as the negation of the existential quantification of (¬p(x) ∨ ¬q(x)). That is, ¬∃x [¬p(x) ∨ ¬q(x)]. Using Premise: ∃x [¬p(x) ∨ ¬q(x)] is not equivalent to ∃x [p(x) ∧ q(x)], and negating it does not produce ∀x [p(x) ∧ q(x)]; rather, ¬∃x [¬p(x) ∨ ¬q(x)] ≡ ∀x ¬[¬p(x) ∨ ¬q(x)] ≡ ∀x [p(x) ∧ q(x)]. This is a different statement from the correct rewrite. Therefore, as stated, the candidate is incorrect for representing ¬∃x [p(x) ∧ q(x)].

In summary:
- The correct rewrite of ¬∃x [p(x) ∧ q(x)] is ∀x [¬p(x) ∨ ¬q(x)].
- The form ¬∃x [¬p(x) ∨ ¬q(x)] would instead correspond to a negation of a different predicate (the negation of the disjunction), and its quantified negation yields ∀x [p(x) ∧ q(x)], not the desired ¬∃x [p(x) ∧ q(x)].

Since there are no answer options provided to evaluate, please share the available choices, and I can analyze each one in the same detailed, step-by-step way.

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