Problem 51 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The proposition (∃x)(∃y)P(x, y) is true because we found a pair of real numbers x and y (x=2, y=1) that satisfy the inequality \(y^2 < x\).

Step 1: Understand the proposition

The proposition (∃x)(∃y)P(x, y) represents the statement "There exists an x and a y such that P(x, y) is true". To determine the truth value of this proposition, we need to find at least one pair of real numbers x and y such that P(x, y) is true.

Step 2: Determine P(x, y) and its conditions

The predicate P(x, y) is represented by the inequality \(y^2 < x\). This means that for the proposition to hold, we need to find values for x and y such that the square of y is less than x. Since both x and y are real numbers, we have a wide range of possibilities to choose from.

Step 3: Find a pair of x and y that satisfy P(x, y)

We can start by examining a simple case to see if it satisfies the predicate. Let's take y = 1, which is a real number. Now, we need to find an x value such that \(1^2 < x\). In this case, \(y^2 = 1\), so we need to find an x value greater than 1. We can choose x = 2, which is also a real number. This gives us the inequality \(1^2 < 2\), which simplifies to \(1 < 2\). This inequality is clearly true. Since we have found a pair of real numbers x and y (x=2, y=1) that satisfy P(x, y), the proposition (∃x)(∃y)P(x, y) is true.

Step 4: Final answer

The proposition (∃x)(∃y)P(x, y) is true because we found a pair of real numbers x and y (x=2, y=1) that satisfy the inequality \(y^2 < x\).

Key Concepts

Predicate Logic

In the realm of mathematics, predicate logic serves as a foundational cognitive tool for expressing propositions involving variables. It extends the classical propositional logic by incorporating predicates and quantifiers. A predicate is essentially a statement that can be true or false depending on the values of its variables.

For instance, in our exercise, \(P(x, y) : y^{2} < x\) is a predicate that involves variables x and y. This predicate becomes either true or false when we assign specific real numbers to x and y. The use of quantifiers like \(\exists\) (which stands for 'there exists') adds depth to our logical expressions, allowing us to formulate statements about some or all elements within a domain of discourse - in this case, the set of real numbers. The proposition \(\exists x)(\exists y) P(x, y)\) declares that there is at least one ordered pair (x, y) in the real numbers that makes the predicate \(P(x, y)\) true.

To properly grasp the concept, one should practice by picking various values for x and y to explore different outcomes and how these outcomes affect the truth value of the statement.

Real Numbers

Real numbers are the bread and butter of mathematical computation and analysis. They include all the numbers on the number line, encompassing whole numbers, integers, rational numbers, and irrational numbers. Real numbers are significant in predicate logic because they provide a spectrum of values we can plug into our predicates.

In our exercise, 'UD' is defined as the set of real numbers which implies that both x and y must be chosen from this continuous, infinite set. The beauty of real numbers is their ability to be as precise as required for any given predicate or equation. Applying real numbers to the predicate \(P(x, y) : y^{2} < x\), we must find real values for x and y that satisfy the inequality. This requires understanding the properties and ordering of real numbers, an essential aspect of solving inequalities.

Inequalities

Inequalities are comparisons between expressions that may not be equal, typically involving '<', '>', '<=', or '>='. Solving inequalities often hinges on understanding the relationships between numbers, particularly when dealing with real numbers.

In the context of our exercise, we encounter the inequality \(y^2 < x\). Here, we must find a pair of real numbers (x, y) such that the square of y is less than x. It's a foundational exercise to not only identify such a pair that fulfils the inequality but also to recognize the infinite number of pairs that would satisfy the condition, as real numbers allow for endless possibilities.

Exercise Improvement Advice

To aid students in grasping inequalities in similar exercises, it's beneficial to demonstrate multiple examples and possibly provide a visual representation of the number line. This could include identifying regions on the graph where the inequality holds true, thus illustrating that the proposition indeed has a truth value when certain pairs are selected from the set of real numbers.