Problem 63 - Discrete Mathematics with Applications

Question

Determine the truth value of each, where \(\mathrm{P}(\mathrm{s})\) denotes an arbitrary predicate. $$(\exists x) P(x) \rightarrow(\exists x) P(x)$$

Step-by-Step Solution

Answer

The given statement is a conditional one with existential quantifiers: \((\exists x) P(x) \rightarrow(\exists x) P(x)\). Since both the antecedent and consequent are the same, the statement is always true regardless of the truth value of P(x). Therefore, the truth value of the statement is TRUE.

Step 1: Understanding the Conditional and Existential Quantifiers

To begin, let's understand the given statement. It is a conditional statement, which has the form \(A \rightarrow B\), where A is the antecedent, and B is the consequent. In this case, both A and B are existential quantifiers, represented by: A: \((\exists x) P(x)\), which means "There exists an x such that P(x) is true." B: \((\exists x) P(x)\), which has the same meaning as A. Now let's analyze the truth value of the statement.

Step 2: Evaluating the Conditional Statement

A conditional statement, \(A \rightarrow B\), is true in all cases except when A is true, and B is false. But in this case, the consequent (B) is the same as the antecedent (A). So, when A is true, B is also true, and the conditional statement is true. Also, when A is false, B is also false, and the conditional statement remains true.

Step 3: Determining the Truth Value

Since all possible cases of the existential quantifiers result in the conditional statement being true, the given statement is always true. Therefore, the truth value of the statement $$(\exists x) P(x) \rightarrow(\exists x) P(x)$$ is TRUE.

Key Concepts

Conditional Statements

Conditional statements are a fundamental aspect of logic. In their basic form, they are structured as "if A, then B" (written as \( A \rightarrow B \)). Here, "A" is known as the antecedent, and "B" is the consequent. In logic, conditional statements are widely used to express scenarios where one event leads to another.Conditional statements have specific truth conditions:

• The statement is false only when the antecedent is true, but the consequent is false.
• In all other scenarios, the conditional statement is considered true.

For example, consider the statement "If it rains, the ground will be wet." This statement would only be false if it rained and the ground remained dry. Otherwise, the statement holds true. Understanding these conditions is crucial for analyzing logical expressions in subjects like mathematics and computer science.

Predicate Logic

Predicate logic extends the ideas of propositional logic by including quantifiers, making it more expressive and powerful. It allows us to make statements about objects and their properties. In predicate logic, we use predicates to assert something about an object. For instance, \( P(x) \) might represent the predicate "x is a cat."There are two primary quantifiers in predicate logic:

• Existential Quantifier (\( \exists \)): This denotes "there exists". For example, \( (\exists x) P(x) \) means "There exists an x such that P(x) is true." In simpler terms, it claims that at least one object makes the proposition true.
• Universal Quantifier (\( \forall \)): This denotes "for all". For instance, \( (\forall x) P(x) \) indicates "For all x, P(x) is true," asserting that the proposition holds for every possible object.

Predicate logic helps us evaluate statements more generally by analyzing relationships and properties of objects within a universe of discourse.

Truth Value Evaluation

Truth value evaluation is a process that determines whether a given logical statement is true or false under specific circumstances. In logic, each statement or proposition has a truth value, either true (T) or false (F).When dealing with complex statements involving quantifiers, such as the one in our example \( (\exists x) P(x) \rightarrow (\exists x) P(x) \), it's crucial to evaluate the truth conditions carefully. Since the antecedent and the consequent are identical, this statement simplifies to examine whether an existence condition implies another that is exactly the same.Let's walk through this example:

• When the antecedent \((\exists x) P(x) \) is true, it means there exists at least one object such that its predicate is true.
• Simultaneously, the consequent \((\exists x) P(x) \) will also be true, because it conveys the same existence condition.

As the conditional statement \( A \rightarrow B \) is true in all scenarios except when the antecedent is true and the consequent is false, and since they are the same here, the overall statement is always true. Understanding how to evaluate truth values accurately is essential for solving logical problems correctly.