Problem 8 - Discrete Mathematics with Applications

Question

Let \(P(x) : x^{2} > x, Q(x) : x^{2}=x,\) and the UD \(=\) set of integers. Determine the truth value of each proposition. $$(\exists x)|\sim P(x)|$$

Step-by-Step Solution

Answer

The truth value of the proposition (\(\exists x\))|\(\sim P(x)\)| is True since there exist integers (0 and 1) that satisfy the inequality \(x^2 \leq x\).

Step 1: P(x) is defined as \(x^2 > x\). The negation of P(x), \(\sim P(x)\), would be the opposite of the original inequality: \(x^2 \leq x\). So, our task is to find an integer x such that \(x^2 \leq x\). #Step 2: Finding an integer for \(\sim P(x)\)#

We will test a few integers to check if any satisfies the inequality \(x^2 \leq x\). 1. For x = 0: \(0^2 \leq 0\) is true, as 0 is equal to 0. 2. For x = 1: \(1^2 \leq 1\) is true, as 1 is equal to 1. 3. For x = 2: \(2^2 \leq 2\) is false, as 4 is greater than 2. 4. For x = -1: \((-1)^2 \leq (-1)\) is false, as 1 is greater than -1. Since we found two integers (0 and 1) that satisfy the inequality \(x^2 \leq x\), we can conclude that the truth value of the proposition (\(\exists x\))|\(\sim P(x)\)| is True.

Step 2: Understand the problem and identify the approach

Read the problem carefully and determine what techniques are needed.

Step 3: Set up and execute the solution

Translate the problem into mathematical expressions and apply techniques.

Step 4: Compute and simplify

Perform calculations and simplify.

Step 5: State the final answer

Present the final answer.

Step 6: Conclude with the answer

The truth value of the proposition (\(\exists x\))|\(\sim P(x)\)| is True since there exist integers (0 and 1) that satisfy the inequality \(x^2 \leq x\).

Key Concepts

Negation of Propositions

In discrete mathematics, understanding the concept of negation is fundamental when analyzing propositions. Negation involves constructing a new proposition, often symbolized by (), which asserts the opposite of what the original proposition claimed. For instance, if we have a proposition stating 'it is raining,' the negation would be 'it is not raining.' The same principle applies to mathematical propositions.

When dealing with inequalities such as in the exercise (), the negation flips the inequality sign. This switch is crucial because it changes what we are claiming about the numbers we are considering. For the inequality (), it is vital to note that the negation turns the greater than '>' sign into a 'less than or equal to' () sign. Consequently, to identify the truth value of the negation, we must find at least one integer that satisfies the new, negated inequality. If such an integer is discovered, the negated proposition holds true within the integer domain.

Quantifiers in Predicate Logic

In predicate logic, quantifiers play a critical role in specifying the nature of the statements we make about our universe of discourse (in this case, the set of integers). There are primarily two types of quantifiers: the existential quantifier (), meaning 'there exists,' and the universal quantifier (), meaning 'for all.'

The existential quantifier is particularly relevant to our problem. When we say (), we announce that there is at least one integer () for which the negated proposition () holds true. It is not a claim that it must be true for all integers, just that one such integer exists. For instance, in the exercise, finding just one integer that satisfies the negation suffices to affirm the truth value of the entire existential statement.

Inequality Solutions in Integer Domain

Solving inequalities within the integer domain can sometimes be more straightforward than within the realm of real numbers, mainly because the set of possibilities is discrete rather than continuous. However, caution is necessary because not all inequalities will have solutions in the integer domain, even if they do in the real numbers.

For the inequality (), we are required to test individual integers to verify if they satisfy the condition. Checking values systematically from a starting point (like 0 or 1) and considering both positive and negative integers is essential. As seen in the step-by-step solution of the exercise, testing integers allows us to conclude affirmatively when we find integer solutions that satisfy the inequality. If at least one integer meets the conditions, as with 0 and 1 in the exercise, the inequality has solutions in the integer domain, and the proposition's truth value is determined accordingly.