Problem 12 - Discrete Mathematics with Applications

Question

Let \(a \in \mathbf{Z} .\) Prove that no integer exists between \(a\) and \(a+1\)

Step-by-Step Solution

Answer

Assuming an integer n between a and a + 1 (\(a < n < a + 1\)), we show it's not possible. Since a and n are integers, n - a is also an integer. But from our assumption, \(0 < n - a < 1\) implies n - a is a positive number less than 1, which is a contradiction, as there's no integer between 0 and 1. Consequently, no integer exists between a and a + 1 for any integer a.

Step 1: Assume an integer between a and a + 1

Let's assume there exists an integer n such that \(a < n < a + 1\), where a and n are integers. Our goal is to show that this assumption is not possible, hence proving that no integer exists between a and a + 1.

Step 2: Use the definition of integer and contradiction

Since a and n are integers, then n - a is also an integer. According to our assumption, \(a < n < a + 1\), which implies that \(0 < n - a < 1\). This means that n - a is a positive number, but less than 1. This is a contradiction because there is no integer between 0 and 1, and we assumed that n - a is an integer.

Step 3: Conclusion

Since our assumption led to a contradiction, we can conclude that there is no integer n between a and a + 1. Therefore, no integer exists between a and a + 1 for any integer a.

Key Concepts

Proof by Contradiction

Proof by contradiction is a fascinating and powerful mathematical technique used to demonstrate the truth of a statement by assuming its opposite. The method reveals that the assumption leads to a contradiction, thereby proving the original assertion is true. Here's how it works:

• We begin with a deliberate assumption that contradicts what we want to prove.
• Under this assumption, we proceed with the logical reasoning.
• If this path leads to a logical impossibility or contradiction, the assumption is false.
• This confirms the original statement is correct.

In the problem we are discussing, we assumed that there exists an integer between two consecutive integers, specifically between \(a\) and \(a+1\). This assumption quickly led us to a contradiction, implying the original statement of no integer existing in between is indeed true. This method is not only applicable to this simple problem but is a foundational technique in proving more complex theorems.

Integers

Integers are one of the foundational concepts in mathematics and are denoted by the symbol \(\mathbf{Z}\). They are a set of numbers that include zero, positive numbers, known as natural numbers, and the negative counterparts of those natural numbers. Here's a breakdown:

• Natural numbers: 1, 2, 3, ...
• Zero: 0
• Negative numbers: -1, -2, -3, ...

One of the key properties of integers is that they are whole numbers, meaning there's no fractional or decimal component. This makes them very rigid in terms of their spacing on the number line. For example, between any two consecutive integers, like \(a\) and \(a+1\), there is no other integer. This property is fundamental in proving that no integer can exist between \(a\) and \(a+1\) in mathematical problems as discussed here.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a given statement based on previously established truths, such as axioms and theorems. In essence, mathematical proofs are the backbone of mathematics as they ensure the reliability and universality of mathematical truths. Here's what defines the essence of a proof:

• It must be logical and free from errors.
• It often begins with known information or premises.
• Through a series of deductive steps, it arrives at a conclusion.
• This conclusion is the statement that has been proven.

In the exercise we explored, a proof by contradiction was employed as a specific type of proof. This logical method confirmed that no integer can exist between \(a\) and \(a+1\). This illustrates how mathematical proofs are crucial for validating hypotheses and ensuring that our understanding of numbers and their properties is sound and consistent.