Suppose the domain of the propositional function P(x, y)...

Suppose the domain of the propositional function $P(x, y)$ consists of pairs $x$ and $y,$ where $x$ is $1,2,$ or 3 and $y$ is $1,2,$ or $3 .$ Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array} $$

Suppose the domain of the propositional function $P(x, y)$ consists of pairs $x$ and $y,$ where $x$ is $1,2,$ or 3 and $y$ is $1,2,$ or $3 .$ Write out these propositions using disjunctions and conjunctions.
$$
\begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array}
$$

Key Concepts

Propositional Function

A propositional function is an expression that contains variables and becomes a proposition—that is, a statement which can be true or false—when the variables are replaced with specific values from its domain. Propositional functions are used to express statements in a generalized form.

Domain of Discourse

The domain of discourse is the set of all objects over which the variables of a propositional function range. It establishes the universe of elements that are considered when evaluating the truth of a propositional function or quantified statement.

Universal Quantifier

The universal quantifier, represented as '?', indicates that a propositional function holds for every element in the domain of discourse. In logical expressions, it often leads to statements that are structured as a large conjunction of instances, one for each element in the domain.

Existential Quantifier

The existential quantifier, denoted as '?', asserts that there is at least one element in the domain of discourse for which the propositional function is true. This is typically represented by a disjunction of instances, ensuring that one or more elements satisfy the property.

Conjunction

Conjunction is the logical operator 'AND' (typically represented by '?') that is used to combine multiple propositions or conditions together in a way that the overall expression is true only when all individual conditions are true. When expanding universally quantified statements over a finite domain, conjunction combines each instance.

Disjunction

Disjunction is the logical operator 'OR' (commonly represented by '?') that connects propositions so that the compound statement is true if at least one of the propositions is true. In the context of existential quantification over a finite domain, disjunction is used to list the instances where the propositional function may be true.

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