Determine the truth value of each of these statements if the...

Determine the truth value of each of these statements if the domain for all variables consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n\left(n^{2} \geq 0\right)} & {\text { b) } \exists n\left(n^{2}=2\right)} \\ {\text { c) } \forall n\left(n^{2} \geq n\right)} & {\text { d) } \exists n\left(n^{2}<0\right)}\end{array} $$

Determine the truth value of each of these statements if the domain for all variables consists of all integers.
$$
\begin{array}{ll}{\text { a) } \forall n\left(n^{2} \geq 0\right)} & {\text { b) } \exists n\left(n^{2}=2\right)} \\ {\text { c) } \forall n\left(n^{2} \geq n\right)} & {\text { d) } \exists n\left(n^{2}<0\right)}\end{array}
$$

Key Concepts

Inequality Analysis in Predicate Logic

Inequality analysis in the context of predicate logic requires systematically testing whether an inequality holds for every element in the domain (for universal quantifiers) or for at least one element (for existential quantifiers). It involves substituting values from the domain into the inequality and logically deducing whether the statement is universally valid or valid for some specific instance.

Domain of Discourse

This concept refers to the set of all objects under consideration when evaluating logical statements. In the context of predicate logic and quantifiers, the domain defines which values the variables can take. For example, when the domain is all integers, every evaluation must use integer values, which can affect the truth or falsity of claims, particularly in equations or inequalities.

Universal Quantifier

The universal quantifier, denoted by '?', indicates that the statement following it must be true for every element in the domain. In practice, this means checking the validity of the proposition for all possible values, which often involves either a logical proof or a counterexample that shows if the proposition fails for any single element.

Existential Quantifier

The existential quantifier, indicated by '?', asserts that there exists at least one element in the domain for which the statement holds true. To validate such statements, it suffices to find a single example within the domain that satisfies the conditions of the statement.

Properties of Integer Squares

This concept involves understanding that when an integer is squared, the result is always a nonnegative number. This property is crucial when evaluating inequalities involving squares, as it immediately implies certain conclusions about the feasibility of statements that involve comparing n² to other numbers.

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