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

Suppose that the domain of the propositional function $P(x)$ consists of the integers $0,1,2,3,$ and $4 .$ Write out each of these propositions using disjunctions, conjunctions, and negations. $$ \begin{array}{llll}{\text { a) }} & {\exists x P(x)} & {\text { b) } \forall x P(x)} & {\text { c) }} \quad {\exists x \neg P(x)} \\ {\text { d) }} & {\forall x \neg P(x)} & {\text { e) } \neg \exists x P(x)} & {\text { f) } \neg \forall x P(x)}\end{array} $$

Suppose that the domain of the propositional function $P(x)$ consists of the integers $0,1,2,3,$ and $4 .$ Write out each of these propositions using disjunctions, conjunctions, and negations.
$$
\begin{array}{llll}{\text { a) }} & {\exists x P(x)} & {\text { b) } \forall x P(x)} & {\text { c) }} \quad {\exists x \neg P(x)} \\ {\text { d) }} & {\forall x \neg P(x)} & {\text { e) } \neg \exists x P(x)} & {\text { f) } \neg \forall x P(x)}\end{array}
$$

Key Concepts

Negation and its Distribution Over Quantifiers

Negation is used to indicate that a statement is not true. When combined with quantifiers, negation interacts with them according to logical equivalences such as De Morgan’s Laws: the negation of an existential quantifier (¬?) is equivalent to a universal quantifier with a negated predicate (?¬), and vice versa (¬? is equivalent to ?¬). This allows for translating complex quantified expressions into logically equivalent forms using basic connectives.

Logical Connectives (Disjunction and Conjunction)

Disjunction (OR) and Conjunction (AND) are the basic logical operations used to combine simpler propositions into more complex expressions. Disjunction represents the idea that at least one of the statements must be true, while conjunction requires that all statements are true simultaneously. In the context of expressing quantified propositions over a finite domain, disjunction is used for existential quantifiers and conjunction for universal quantifiers.

Existential Quantifier

The existential quantifier, denoted by ?, asserts that at least one element in a given domain makes the propositional function true. When the domain is finite, an expression such as ?x P(x) can be written as a series of disjunctions (OR) between propositions for each element in the domain.

Universal Quantifier

The universal quantifier, denoted by ?, states that every element in the domain makes the propositional function true. For a finite domain, a universal statement such as ?x P(x) can be expressed as a series of conjunctions (AND) of propositions for each element of the domain.

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