Let Q(x, y) be the statement "x x+y=x-y ." If the domain for...

Let $Q(x, y)$ be the statement "x $x+y=x-y$ ." If the domain for both variables consists of all integers, what are the truth values? $$\begin{array}{ll}{\text { a) } Q(1,1)} & {\text { b) } Q(2,0)} \\ {\text { c) } \forall y Q(1, y)} & {\text { d) } \exists x Q(x, 2)} \\{\text { e) } \exists x \exists y Q(x, y)} & {\text { f) } \forall x \exists y Q(x, y)} \\ {\text { g) } \exists y \forall x Q(x, y)} & {\text { h) } \forall y \exists x Q(x, y)} \\ {\text { i) } \forall x \forall y Q(x, y)}\end{array}$$

Let $Q(x, y)$ be the statement "x $x+y=x-y$ ." If the domain for both variables consists of all integers, what are the truth values?
$$\begin{array}{ll}{\text { a) } Q(1,1)} & {\text { b) } Q(2,0)} \\ {\text { c) } \forall y Q(1, y)} & {\text { d) } \exists x Q(x, 2)} \\{\text { e) } \exists x \exists y Q(x, y)} & {\text { f) } \forall x \exists y Q(x, y)} \\ {\text { g) } \exists y \forall x Q(x, y)} & {\text { h) } \forall y \exists x Q(x, y)} \\ {\text { i) } \forall x \forall y Q(x, y)}\end{array}$$

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Key Concepts

Predicate Logic

Predicate logic extends propositional logic by incorporating variables and predicates to express statements about elements in a domain. It provides a framework to discuss properties of and relations between objects.

Quantifiers

Quantifiers, including the universal quantifier (?) and the existential quantifier (?), specify the scope of a statement within a domain. They determine whether a condition must hold for every element or if it is sufficient for it to hold for at least one element.

Domain of Discourse

The domain of discourse is the set of all objects considered in a logical analysis. It defines the universe over which the variables range and is crucial in determining the truth values of statements involving quantifiers.

Truth Value Evaluation

Truth value evaluation in predicate logic requires checking whether a given predicate holds when variables are replaced by elements from the domain. This process involves substituting values for free and bound variables according to the structure of the logical statement, especially in the presence of quantifiers.

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