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. $$ \begin{array}{ll}{\text { a) } \forall n \exists m\left(n^{2}<m\right)} & {\text { b) } \exists n \forall m\left(n<m^{2}\right)} \\ {\text { c) } \forall n \exists m(n+m=0)} & {\text { d) } \exists n \forall m(n m=m)} \\ {\text { e) } \exists n \exists m\left(n^{2}+m^{2}=5\right)} & {\text { f) } \exists n \exists m\left(n^{2}+m^{2}=6\right)}\end{array} $$ $$ \begin{array}{l}{\text { g) } \exists n \exists m(n+m=4 \wedge n-m=1)} \\ {\text { h) } \exists n \exists m(n+m=4 \wedge n-m=2)} \\ {\text { i) } \forall n \forall m \exists p(p=(m+n) / 2)}\end{array} $$

Determine the truth value of each of these statements if the domain for all variables consists of all integers.
$$
\begin{array}{ll}{\text { a) } \forall n \exists m\left(n^{2}<m\right)} & {\text { b) } \exists n \forall m\left(n<m^{2}\right)} \\ {\text { c) } \forall n \exists m(n+m=0)} & {\text { d) } \exists n \forall m(n m=m)} \\ {\text { e) } \exists n \exists m\left(n^{2}+m^{2}=5\right)} & {\text { f) } \exists n \exists m\left(n^{2}+m^{2}=6\right)}\end{array}
$$
$$
\begin{array}{l}{\text { g) } \exists n \exists m(n+m=4 \wedge n-m=1)} \\ {\text { h) } \exists n \exists m(n+m=4 \wedge n-m=2)} \\ {\text { i) } \forall n \forall m \exists p(p=(m+n) / 2)}\end{array}
$$

Key Concepts

Predicate Logic

Predicate logic is a formal system in mathematics and logic that uses quantifiers and predicates to express statements about objects in a domain. It extends propositional logic by allowing the use of quantifiers and variables, enabling more expressive statements about the relationships and properties of objects.

Universal Quantifier

The universal quantifier, denoted by ?, is used to state that a certain property or relation holds for all elements in the domain. Evaluating statements with a universal quantifier involves checking that every possible element in the domain satisfies the given condition.

Existential Quantifier

The existential quantifier, denoted by ?, asserts that there is at least one element in the domain for which a given property or relation holds. In evaluating such statements, it is sufficient to find one element that satisfies the condition for the statement to be true.

Domain of Discourse

The domain of discourse is the set of objects over which the variables in a logical statement range. In the context of many logical problems, specifying the domain is crucial because the truth value of the quantifiers and the overall statement can depend on the properties of the domain, such as the set of integers.

Truth Value Evaluation

Determining the truth value of a logical statement involves systematically analyzing the statements defined by the quantifiers and predicates over the specified domain. This process may require constructing examples or counterexamples, as well as applying known properties of the domain elements.

Order of Quantifiers

The order of quantifiers in a logical expression is significant because it determines the structure of the statement and how the variables interact. Nested quantifiers, especially when they alternate between universal and existential, often require careful interpretation to ascertain the overall meaning and truth value of the statement.

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