Suppose that the domain of the propositional function P(x)...

Suppose that the domain of the propositional function $P(x)$ consists of the integers $1,2,3,4,$ and $5 .$ Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. $$ \begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \neg \exists x P(x)} & {\text { d) } \neg \forall x P(x)}\end{array} $$ e) $\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)$

Suppose that the domain of the propositional function $P(x)$ consists of the integers $1,2,3,4,$ and $5 .$ Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.
$$
\begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \neg \exists x P(x)} & {\text { d) } \neg \forall x P(x)}\end{array}
$$
e) $\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)$

Key Concepts

Negation and De Morgan's Laws

Negations of quantified statements can be translated using De Morgan's laws along with the quantifier conversions. For instance, the negation of an existential statement ¬?x P(x) is equivalent to stating that the predicate does not hold for any element, translating to ¬P(1) ? ¬P(2) ? ¬P(3) ? ¬P(4) ? ¬P(5). Similarly, ¬?x P(x) translates to a disjunction with at least one instance where P does not hold.

Universal Quantifier

The universal quantifier states that the predicate holds for every element in the domain. In a finite setting, this is converted into a conjunction of the predicate for each element, for instance, P(1) ? P(2) ? P(3) ? P(4) ? P(5).

Implication Transformation

Implications in logical formulas are often rewritten in terms of negations and disjunctions. The equivalence a ? b ? ¬a ? b is used to remove the implication. For example, a conditional statement such as (x ? 3) ? P(x) can be rewritten as (x = 3) ? P(x). This conversion is essential for expressing statements without the implication operator, especially when expanding the formula over a finite domain.

Finite Domain Expansion

When the domain is finite, quantified statements can be rewritten by explicitly listing all instances in the domain. For example, an existential quantifier is replaced by a disjunction of the statements for each element, and a universal quantifier by a conjunction across all elements. This technique removes the need for quantifiers in the logical formula.

Existential Quantifier

The existential quantifier asserts that there is at least one element in the domain for which the predicate holds. In a finite domain like {1, 2, 3, 4, 5}, this is expressed using a disjunction of the predicate applied to each element, such as P(1) ? P(2) ? P(3) ? P(4) ? P(5).

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