Problem 48 - Discrete Mathematics with Applications

Question

Let UD \(=\) set of integers, \(\mathrm{P}(x, y) : x\) is a multiple of \(y,\) and \(Q(x, y) : x \geq y\) Determine the truth value of each proposition. $$(\forall x) | P(x, 3) \rightarrow Q(x, 3) ]$$

Step-by-Step Solution

Answer

The truth value of the proposition \((\forall x)[P(x, 3) \rightarrow Q(x, 3)]\) is False, as it doesn't hold true for all integers x (it fails for negative multiples of 3).

Step 1: Define the relevant predicates and proposition

The given predicates are P(x, y) : x is a multiple of y, and Q(x, y) : x ≥ y. We are asked to determine the truth value of the proposition: \((\forall x)[P(x, 3) \rightarrow Q(x, 3)]\)

Step 2: Analyze the implication in the proposition

The proposition is of the form P(x, 3) → Q(x, 3). This can be elaborated as: If x is a multiple of 3, then x is greater than or equal to 3. We need to check if this statement is true for all integers x.

Step 3: Check the statement for integer values of x

Let's systematically investigate some integer cases of x. - For positive multiples of 3, say x = 3, 6, 9,..., since it's a multiple of 3, P(x, 3) is true. And also x ≥ 3, so Q(x, 3) is true. Therefore, P(x, 3) → Q(x, 3) holds true. - For negative multiples of 3, say x = -3, -6, -9,..., since it's a multiple of 3, P(x, 3) is true. But x is not greater than or equal to 3, so Q(x, 3) is false. Therefore, P(x, 3) → Q(x, 3) does not hold. - For non-multiples of 3, say x = 1, 2, 4, 5,..., P(x, 3) is false. In this case, regardless of the value of Q(x, 3), the implication P(x, 3) → Q(x, 3) is true. From these observations, we see that the implication in the proposition does not hold true for negative multiples of 3.

Step 4: Determine the truth value of the proposition

Since the proposition (∀x)[P(x, 3) → Q(x, 3)] doesn't hold true for all integers x (it fails for negative multiples of 3), the truth value of the proposition is: False

Key Concepts

Universal Quantification

In predicate logic, universal quantification involves statements that claim something for all elements of a certain category. The symbol \( \forall \) is used to denote "for all" or "for every." For example, in the proposition \((\forall x)[P(x, 3) \rightarrow Q(x, 3)]\), we express that for every integer \(x\), if \(x\) is a multiple of 3, then \(x\) is greater than or equal to 3.

This implies a sweeping generalization. It creates a blanket claim that needs to be true for every possible value in the domain of all integers. In practice, when we attempt to determine if such a statement is true, we examine each case, covering all typical instances of the condition. If even one case fails, the whole statement is false. This characteristic makes universal quantification stringent and absolute.

Implications in Logic

Implications in logic, denoted by \( \rightarrow \), express a relationship where if one statement (the antecedent) is true, then another (the consequent) follows. The conditional statement "if-then" can be confusing but follows specific logical rules. In our specific context, \(P(x, 3) \rightarrow Q(x, 3)\) can be unpacked as:

• If \(x\) is a multiple of 3, then \(x\) is greater than or equal to 3.

Interesting properties of implications include:

• If the antecedent \(P(x, 3)\) is false, the implication is automatically true, regardless of \(Q(x, 3)\).
• If the antecedent \(P(x, 3)\) is true and the consequent \(Q(x, 3)\) is false, the implication is false.
• In all other cases, the implication holds true.

Understanding these rules is key to analyzing the truth values of logical statements. They dictate how propositions are evaluated in formal logic.

Truth Value Determination

Determining the truth value of a proposition requires careful analysis of each possible case, especially when the domain includes infinite elements such as integers. In the statement \((\forall x)[P(x, 3) \rightarrow Q(x, 3)]\), the truth value hinges on verifying the implication for each integer \(x\).

We categorize \(x\) into three groups for ease of assessment:

• **Positive Multiples of 3**: Here both conditions of the implication hold, since any positive multiple of 3 is also greater than or equal to 3.
• **Negative Multiples of 3**: These fail the consequent \(Q(x, 3)\) since such numbers are less than 3, leading to a false implication.
• **Non-Multiples of 3**: For these values, the antecedent \(P(x, 3)\) is false, so the implication is true regardless of the consequent.

Since at least one subset, namely negative multiples of 3, leads to a false statement, the universal proposition itself is false. This example showcases why checking all potential scenarios is crucial in logic, reaffirming the dependency of truth values on specific instances within the set of integers.