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

Determine the truth value of each of these statements if the domain consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \quad \exists x\left(x^{3}=-1\right)} & {\text { b) } \exists x\left(x^{4}<x^{2}\right)} \\ {\text { c) } \quad \forall x\left((-x)^{2}=x^{2}\right)} & {\text { d) } \forall x(2 x>x)}\end{array} $$

Determine the truth value of each of these statements if the domain consists of all real numbers.
$$
\begin{array}{ll}{\text { a) } \quad \exists x\left(x^{3}=-1\right)} & {\text { b) } \exists x\left(x^{4}<x^{2}\right)} \\ {\text { c) } \quad \forall x\left((-x)^{2}=x^{2}\right)} & {\text { d) } \forall x(2 x>x)}\end{array}
$$

Key Concepts

Domain of Discourse

In predicate logic, the domain of discourse specifies the set of all objects over which variables can range. In this case, the domain is the set of all real numbers, meaning that every variable in the statements is assumed to be a real number. This influences how equations and inequalities are evaluated since properties specific to real numbers are applicable.

Existential Quantifier

The existential quantifier, denoted by ?, is used to state that there is at least one element in the domain that makes the predicate true. When a statement begins with ?, such as ? x (x³ = -1) or ? x (x? < x²), it asserts the existence of a real number for which the given equation or inequality holds.

Universal Quantifier

The universal quantifier, represented by ?, asserts that a predicate holds for every element in the domain. When a statement is phrased as ? x, it requires that the property or equation must be true for all real numbers. This is critical for testing claims like ? x ((-x)² = x²) or ? x (2x > x).

Polynomial Equations and Inequalities

Polynomial equations and inequalities refer to expressions involving variables raised to whole-number powers. Solving polynomial equations like x³ = -1 or analyzing inequalities such as x? < x² involves finding specific values or intervals within the domain where the equality or inequality holds. This typically requires finding roots, considering sign changes, and applying properties of exponents.

Properties of Even Functions and Squaring

The property that (-x)² equals x² for all real numbers is an example of a function demonstrating even symmetry, where the function's value is unchanged when x is replaced with -x. This is due to the squaring operation eliminating the sign of the original number, which is an important property when analyzing expressions that involve even powers of variables.

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