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Real numbers, denoted as \(\mathbf{R}\), are numbers that include both rational numbers like fractions, and irrational numbers that cannot be expressed as simple fractions. They encompass a wide variety of numbers such as positive numbers, negative numbers, and zero. Real numbers can be found on the number line, providing a comprehensive way to represent different quantities in the real world.

• Rational numbers: Numbers like \(\frac{1}{2}, -3,\) and \(5\) are rational because they can be expressed as a fraction of two integers.
• Irrational numbers: Numbers such as \(\sqrt{2}\) and \(\pi\) that cannot be written as fractions are irrational.
• Negative numbers: Numbers less than zero, which are found on the left side of the number line, like \(-2\).
• Positive numbers: Numbers greater than zero, such as \(1\), \(5.5\), which are found on the right side of the number line.

An inequality is a mathematical statement that indicates that one value is larger or smaller than another. Inequalities are used to show the relative size of two values. In the predicate logic problem, inequalities play a crucial role.For instance, in the predicate \(Q(x,y)\), the statement "If \(x < y\)" involves an inequality. Here are some basics:

• 'Less than' (\(<\)) indicates that one value is smaller than the other.
• 'Greater than' (\(>\)) means that one value is larger than another.
• 'Less than or equal to' (\(\leq\)) and 'greater than or equal to' (\(\geq\)) broaden the basic inequalities to include equality.

Understanding these basic inequalities and how they are used is critical for setting up and solving mathematical problems, particularly when dealing with predicate logic.

A mathematical proof is a logical process to demonstrate the truth or falsity of a given statement. The foundation of mathematical reasoning, proofs are used to substantiate claims and assertions within mathematics.In the given exercise, proving or disproving the statement \(Q(x,y)\) involves evaluating whether the premise leads to the conclusion.

• Assumption: Begin with the initial condition or what is assumed to be true, such as \(x < y\).
• Evaluation: Check subsequent steps (e.g., whether \(x^2 < y^{2n}\)).
• Conclusion: Decide if the original predicate statement holds true or false based on evaluations.

Proofs often require understanding the properties and operations involving real numbers and inequalities to form concrete conclusions about mathematical statements.

In mathematics, the domain and range are essential concepts used to describe the set of possible inputs and their corresponding outputs for a function or relation. - **Domain**: This is the complete set of possible values of the independent variable; in context, it describes all possible values \(x\) and \(y\) can take in the problem, which are real numbers (\(\mathbf{R}\)). This means any real number is a potential input for \(x\) and \(y\). - **Range**: This refers to the set of possible output values after applying a rule of function or relation, such as squaring \(x\) and raising \(y\) to a power in \(y^{2n}\).Understanding these helps in determining which values satisfy all parts of the predicate \(Q(x,y)\), as we need to verify the statement holds over the entire domain of real numbers.