Problem 56 - Discrete Mathematics with Applications

Question

A third useful quantifier is the uniqueness quantifier \(\exists ! .\) The proposition \(\left(\exists^{\prime} x\right) P(x)\) means There exists a unique (meaning exactly one) \(x\) such that \(P(x) .\) Determine the truth value of each proposition, where UD \(=\) set of integers. $$\left(\exists^{\prime} x\right)(x+3=3)$$

Step-by-Step Solution

Answer

The truth value of the proposition \((\exists x)(x+3=3)\), where the Universal Domain (UD) is a set of integers, is True. This is because there exists a unique integer solution for \(x\), which is \(x=0\).

Step 1: Analyzing the equation

Given the equation \(x+3=3\), our objective is to find if there exists a unique integer solution for \(x\).

Step 2: Solving the equation

To find the solution for \(x\), we can subtract 3 from both sides of the equation to isolate \(x\). \(x+3-3 = 3-3\) This simplifies to: \(x=0\)

Step 3: Verifying the solution

Now that we have found a solution for \(x\), we need to verify if it is unique and lies within the Universal Domain (set of integers). 1. Check if the solution is an integer: Indeed, 0 is an integer. 2. Check for the uniqueness: Since there is no other integer value that can satisfy the given equation, we can say that the solution is unique.

Step 4: Determine the truth value

As there exists a unique integer value (exactly one) that satisfies the given equation, we can conclude that the proposition \((\exists x)(x+3=3)\) is true. So, the truth value of the proposition is True.

Key Concepts

Discrete Mathematics and the Uniqueness Quantifier

Discrete mathematics is a branch of mathematics that deals with discrete elements that are distinct and separate values. This often involves study of integers, graphs, and statements in logic. Within this domain, the concept of a uniqueness quantifier plays a critical role. It's a tool that allows mathematicians to express the idea that there is exactly one solution to a particular problem or proposition.

In our exercise, we encounter the uniqueness quantifier represented by \( \exists! \), which is read as 'there exists exactly one'. This quantifier is used to affirm that a statement holds true for one and only one element in the universal set, in this case, the set of integers. So when we evaluate the equation \(x + 3 = 3\), we’re being asked to determine whether there is a single integer solution for \(x\) that makes the equation true. By solving, we isolate \(x\) which gives us the value of zero, demonstrating a unique solution within our defined universal set. Therefore, in discrete mathematics, specifying uniqueness is vital because it distinguishes between having at least one solution and having one and only one solution, which fundamentally changes the meaning of a proposition.

Truth Value in Logical Propositions

The truth value of a proposition in logic is a determination of whether a statement is true or false. Each proposition in logic is assigned a truth value. In the context of the given exercise, we analyze the truth value of a statement by assessing whether the uniqueness quantified proposition is satisfied.

By using the uniqueness quantifier, we declare that there should be a single integer value that resolves the equation. After finding that \(x = 0\) is the unique solution, we must evaluate whether this solution is the only one within the set of integers. Since we can confirm that zero is indeed the only integer that satisfies \(x + 3 = 3\), we conclude that the proposition's truth value is True. Truth values are the cornerstone of logical reasoning in discrete mathematics, as they allow us to make precise determinations about the validity of mathematical statements.

Integer Solutions to Algebraic Equations

When dealing with algebraic equations within discrete mathematics, an integer solution is a solution that results in an integer value. Integer solutions are integral to understanding mathematical problems that strictly involve whole numbers.

In our example, the equation \(x+3=3\) is analyzed for integer solutions. Integer solutions are important as they can articulate the feasibility of scenarios in number theory, cryptography, and computer science where only whole numbers are applicable. After simplifying the equation by subtracting 3 from both sides, the clear solution is \(x=0\), which is a valid integer. This highlights the significance of checking not only for solutions but for solutions that adhere to the confines of the problem's domain. In this case, determining that the solution is a member of the set of integers verifies that we have a correct and relevant solution to our equation.