Let Q(x, y) be the statement "x has sent an e-mail message...

Let $Q(x, y)$ be the statement "x has sent an e-mail message to $y,$ " where the domain for both $x$ and $y$ consists of all students in your class. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array} $$

Let $Q(x, y)$ be the statement "x has sent an e-mail message to $y,$ " where the domain for both $x$ and $y$ consists of all students in your class. Express each of these quantifications in English.
$$
\begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array}
$$

Key Concepts

Predicate Logic

Predicate logic is a formal system in which statements are expressed using predicates, which are functions that return a truth value, and quantifiers, which specify the extent to which a predicate applies over a set or domain. It allows the representation of relations among objects using logical formulas.

Existential Quantifier

The existential quantifier, denoted by '?', is used to assert that there is at least one element in the domain for which the predicate holds true. It is a way of stating that something exists within the given context or domain that satisfies a particular property.

Universal Quantifier

The universal quantifier, denoted by '?', is used to assert that a predicate holds true for all elements in the domain. It is a way of expressing that every member of the domain satisfies a specific condition or property.

Order of Quantifiers

The order of quantifiers in a logical formula is significant because it determines the scope and interaction of the existential and universal claims being made. Changing the order can lead to different interpretations: for example, the statement 'there exists an x such that for all y, Q(x, y)' is not equivalent to 'for all y, there exists an x such that Q(x, y)'.

Domain of Discourse

The domain of discourse refers to the set of all objects over which the quantifiers in a logical formula range. It provides the context within which the properties or relations described by the predicates are evaluated, ensuring that the logical statements have a clear and well-defined scope.

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