Suppose that the domain of Q(x, y, z) consists of triples x,...

Suppose that the domain of $Q(x, y, z)$ consists of triples $x, y, z,$ where $x=0,1,$ or $2, y=0$ or $1,$ and $z=0$ or $1 .$ Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall y Q(0, y, 0)} & {\text { b) } \exists x Q(x, 1,1)} \\ {\text { c) } \exists z \neg Q(0,0, z)} & {\text { d) } \exists x \neg Q(x, 0,1)}\end{array} $$

Suppose that the domain of $Q(x, y, z)$ consists of triples $x, y, z,$ where $x=0,1,$ or $2, y=0$ or $1,$ and $z=0$ or $1 .$ Write out these propositions using disjunctions and conjunctions.
$$
\begin{array}{ll}{\text { a) } \forall y Q(0, y, 0)} & {\text { b) } \exists x Q(x, 1,1)} \\ {\text { c) } \exists z \neg Q(0,0, z)} & {\text { d) } \exists x \neg Q(x, 0,1)}\end{array}
$$

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Key Concepts

Finite Domain Expansion

When working in predicate logic with a finite domain, quantifiers (universal or existential) can be eliminated by explicitly listing all elements in the domain. This technique allows one to rewrite quantified statements as combinations of simpler logical expressions (conjunctions for universals and disjunctions for existentials) over the specific, finite set of elements where the predicate is evaluated.

Universal Quantifier Expansion

A universally quantified statement, such as ?x P(x), over a finite domain can be rewritten as a conjunction of the predicate applied to each element in the domain. This is because the statement must hold true for every element, so if even one instance fails, the overall statement is false. This process makes it clear how the universal condition is distributed across all elements.

Existential Quantifier Expansion

An existentially quantified statement, such as ?x P(x), over a finite domain can be rewritten as a disjunction of the predicate applied to each element in the domain. This reflects that for the overall statement to be true, it is sufficient for the predicate to be true when applied to at least one member of the domain. The rewriting clarifies that the existence of a single satisfying instance is enough.

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