给定论域的命题函数量词表达式转非量词逻辑式的求解问询

Let’s break this down step by step to get the right result, using only negation, disjunction (∨), and conjunction (∧):

Step 1: Expand the existential quantifier $\exists x(\neg P(x))$ #

Existential quantifiers translate to disjunctions (OR operations) across the entire domain—we just need at least one element where $\neg P(x)$ is true. For our domain ${-5, -3, -1, 1, 3, 5}$, this expands directly to:
¬P(-5) ∨ ¬P(-3) ∨ ¬P(-1) ∨ ¬P(1) ∨ ¬P(3) ∨ ¬P(5)

Step 2: Expand the universal quantifier $\forall x((x < 0) → P(x))$ #

First, recall that the implication $A → B$ is logically equivalent to $\neg A ∨ B$. So $(x < 0) → P(x)$ rewrites to $\neg(x < 0) ∨ P(x)$ (or equivalently, $(x ≥ 0) ∨ P(x)$).

Universal quantifiers translate to conjunctions (AND operations) across every element in the domain. Let’s evaluate this for each x in our set:

• For $x = -5, -3, -1$ (all values less than 0): $\neg(x < 0)$ is false, so $\neg(x < 0) ∨ P(x)$ simplifies to just $P(x)$ (since false OR any statement equals that statement).
• For $x = 1, 3, 5$ (all values ≥ 0): $\neg(x < 0)$ is true, so $\neg(x < 0) ∨ P(x)$ simplifies to true (since true OR any statement is always true).

When we take the conjunction of all these results, the "true" terms don’t affect the outcome (ANDing with true leaves the rest of the statement unchanged). So the universal quantifier part simplifies to:
P(-5) ∧ P(-3) ∧ P(-1)

Step 3: Combine and simplify the two parts #

Now we just combine the two expanded expressions with a conjunction (since the original statement is $\exists x(\neg P(x)) \land \forall x((x < 0) → P(x))$):
(¬P(-5) ∨ ¬P(-3) ∨ ¬P(-1) ∨ ¬P(1) ∨ ¬P(3) ∨ ¬P(5)) ∧ (P(-5) ∧ P(-3) ∧ P(-1))

We can simplify this further by noticing that $\neg P(-5)$ can’t be true if $P(-5)$ is true (and the same applies to -3 and -1). This means the only way the entire statement holds is if all negative x satisfy P(x), and at least one non-negative x does NOT satisfy P(x). The simplified version is:
(P(-5) ∧ P(-3) ∧ P(-1)) ∧ (¬P(1) ∨ ¬P(3) ∨ ¬P(5))

Both forms are logically correct, but the simplified version is more concise and makes the underlying logic easier to read.

内容的提问来源于stack exchange，提问作者ahmelq