Problem 52 - Discrete Mathematics with Applications

Question

Let UD = set of real numbers and \(P(x, y): y^{2} < x .\) Determine the truth value of each proposition. $$(\forall x)(\exists y) \mathrm{P}(x, y)$$

Step-by-Step Solution

Answer

The truth value of \((\forall x)(\exists y) \mathrm{P}(x, y)\), where \(P(x, y) = y^{2} < x\), is false. This is because there are cases (x = 0, and x < 0) where we cannot find a suitable y value satisfying the inequality.

Step 1: Understand Quantifiers and Predicate

We are given the predicate P(x, y): \(y^{2} < x\), where x and y are both real numbers. In this case, there are two quantifiers: - \((\forall x)\) means "for all x", where x is in the set of real numbers (UD). - \((\exists y)\) means "there exists a y", where y is also in the set of real numbers (UD). The proposition \((\forall x)(\exists y) \mathrm{P}(x, y)\) is read as "For all x, there exists a y such that \(y^{2} < x\)". Now, our task is to determine the truth value of this proposition.

Step 2: Analyze the Predicate with Respect to Quantifiers

We need to consider each value of x in the set of real numbers and determine if there exists a corresponding value of y that satisfies the inequality \(y^{2} < x\). If this holds true for every x, then the proposition is true; otherwise, it is false.

Step 3: Evaluate the Proposition

Let's analyze the inequality \(y^{2} < x\) for different x values in UD. 1. If x > 0, we can find a y such that \(y^{2} < x\). Simply choose y = 0, and it is clear that \(0^{2} < x\). 2. If x = 0, we cannot find a real number y such that \(y^{2} < 0\), since the square of any real number is always non-negative. 3. If x < 0, it is also impossible to find a real number y such that \(y^{2} < x\), for the same reason as mentioned above. Since we have found cases (x = 0, and x < 0) where we cannot find a suitable y value satisfying the inequality, the proposition is false. Therefore, the truth value of \((\forall x)(\exists y) \mathrm{P}(x, y)\) is false.

Key Concepts

Quantifiers

In predicate logic, quantifiers play a crucial role in understanding mathematical statements. They help specify how a property or condition is applied to elements within a particular domain, which in this case, is the set of real numbers.

• The universal quantifier \((\forall x)\) signifies that the statement must hold true for every element \(x\) in the domain.
• The existential quantifier \((\exists y)\) indicates that there exists at least one element \(y\) within the domain for which the statement is true.

Understanding the combination of quantifiers is vital. For example, in the proposition \((\forall x)(\exists y) P(x, y)\), it means "for every \(x\), there is at least one \(y\) such that the condition holds." The order of quantifiers impacts the interpretation and outcome of logical statements significantly.

Real Numbers

Real numbers form the backbone of many mathematical analyses and encompass all the numbers we commonly use. They include:

• Rational numbers: numbers that can be expressed as a fraction, such as \(\frac{3}{4}\) or -5.
• Irrational numbers: numbers that cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \(\pi\).
• Integers: whole numbers, including negatives, zero, and positive numbers.

The set of real numbers is denoted by \(\mathbb{R}\), and it is continuous, meaning there is no "next" real number. Understanding real numbers is essential, as in the given logical proposition, both variables \(x\) and \(y\) are considered within this set. This means that the analysis and determination of truth values involve considering the limitless possibilities within the real number span.

Truth Value

The truth value of a proposition indicates whether the proposition is true or false in a given context. To find the truth value of \((\forall x)(\exists y) P(x, y)\), we need to assess whether the inequality \(y^2 < x\) holds for all real \(x\).Analysis shows:

• For \(x > 0\), choosing \(y = 0\) easily satisfies \(y^2 < x\).
• For \(x = 0\), no real \(y\) satisfies \(y^2 < 0\), because squared values are non-negative.
• For \(x < 0\), it's impossible for any real \(y\) to satisfy \(y^2 < x\) since \(y^2\) remains non-negative, and thus can't be smaller than a negative number.

These observations help conclude that the proposition is false, as there are specific instances where the proposition fails to hold true. Thus, its truth value is "false."

Inequality Analysis

Inequalities like \(y^2 < x\) are fundamental in analyzing logical propositions. Here, analyzing \(y^2 < x\) involves understanding how the inequality applies to real numbers.

• When \(x > 0\), the choice of \(y = 0\) means \(0^2 < x\) holds true, demonstrating that solutions exist for positive \(x\).
• When \(x = 0\), no choice of real \(y\) will satisfy \(y^2 < 0\), since squares of real numbers are always \(\geq 0\).
• For \(x < 0\), any possibility of \(y^2 < x\) is thwarted, for \(y^2\) is non-negative and cannot be less than any negative \(x\).

Such analyses are pivotal in logic, showing when and how solutions exist and illuminating the limitations of the inequality within the context of the proposition.