Problem 61 - 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. $$(\forall x)(\exists ! y)(x+y=4)$$

Step-by-Step Solution

Answer

The given proposition is \((\forall x)(\exists ! y)(x+y=4)\), which means "for every integer x, there exists a unique integer y such that x + y = 4." By choosing any integer x and solving for y with the equation \(y = 4 - x\), we can verify that there is a unique value for y for any chosen integer x. Therefore, the proposition is true.

Step 1: Understand the proposition

The proposition we are given is: \((\forall x)(\exists ! y)(x+y=4)\) which means "for every integer x, there exists a unique integer y such that x + y = 4."

Step 2: Choose any integer x and find y

To determine if the statement is true, we can choose any integer x and solve for y. The equation given is \(x + y = 4\). To find \(y\), we can write: \(y = 4 - x\)

Step 3: Test the uniqueness of y

Now we'll test whether there is a unique value for y that makes the equation true for any given x in the set of integers. Recall the equation for y: \(y = 4 - x\) For a chosen value of x, there is only one possible value for y that will make the equation true. So, the uniqueness condition is satisfied. Since there is a unique value for y for any chosen integer x, the proposition \((\forall x)(\exists ! y)(x+y=4)\) is true.

Key Concepts

Understanding Propositions

A proposition is a statement that can be clearly identified as either true or false, but not both simultaneously. In mathematics, propositions are often used to make formal statements about mathematical truths. For instance, the example provided, \(\forall x)(\exists ! y)(x+y=4)\), is a proposition that asserts a specific relationship between two variables, \(x\) and \(y\), within the set of integers. Furthermore, propositions can often contain quantifiers like \(\forall\) (meaning 'for all') and \(\exists !\) (indicating 'there exists a unique'), which determine the scope and nature of the variables involved. As propositions involve clearly outlined conditions, improving comprehension involves not only clarifying their structure and components but also understanding the logical operations and algebraic manipulations that can be employed to evaluate their truth value.

Evaluating Propositions

When evaluating the truth value of a proposition, begin by translating the symbolic statement into plain language. Once translated, consider the set over which the variables are defined and apply the necessary operations to discern the truth value. It's critical to verify any conditions specified by the proposition, such as uniqueness, to properly assess its validity.

The Set of Integers

The set of integers, represented by the symbol \(\mathbb{Z}\), includes all the whole numbers and their negatives: \( ..., -3, -2, -1, 0, 1, 2, 3, ...\) It's important to understand that the set of integers is infinite and includes no fractions or decimal numbers. In exercises involving integer sets, computations are typically straightforward, but one should take care to distinguish integers from other number sets.

Relevance to Propositions

When a proposition such as \(\left(\exists^\prime x\right) \mathrm{P}(x)\) is stated with the domain being the set of integers, it directly affects the potential values that the variables \(x\) and \(y\) can assume. Therefore, when the exercise states that \(\mathrm{UD}=\) set of integers, it's signaling that \(x\) and \(y\) should be treated as integers, a constraint that factors into finding and verifying a unique solution.

Truth Value in Mathematics

The truth value of a proposition refers to its validity within a logical system or set context. It's binary: a proposition can either be true or false. This black and white evaluation is crucial in mathematical proof and reasoning.

Determining Truth Values

To determine the truth value of a proposition like \(\left(\exists^\prime x\right) \mathrm{P}(x)\), follow the defined conditions and logic. In our example, for each integer \(x\), we seek an integer \(y\) such that the equation \(x+y=4\) holds true with a single unique solution for \(y\). The process demonstrated in the solution, considering any \(x\) from the integers and showing that a unique \(y\) necessarily follows, concludes that the proposition holds a true truth value across the entire set of integers. This unambiguity in solution is pivotal for the proposition to be considered true. When studying, always ensure the unique conditions are fulfilled to properly evaluate truth value.