Problem 56 - Discrete Mathematics with Applications

Question

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}\) $$\left[\frac{n^{2}}{4}\right]=\frac{n^{2}+3}{4} \text { if } n \text { is odd }$$

Step-by-Step Solution

Answer

Given that \(n\) is odd, we can express it as \(n=2k+1\). The left-hand side of the equation \(\left[\frac{n^{2}}{4}\right] = \left[\frac{(2k+1)^{2}}{4}\right] = \left[\frac{4k^2 + 4k + 1}{4}\right] = k^2+k\). The right-hand side of the equation \(\frac{n^{2}+3}{4} = \frac{(2k+1)^{2}+3}{4} = \frac{4k^2 + 4k + 1 + 3}{4} = k^2+k\). Since both sides are equal, the statement is proven: \(\left[\frac{n^{2}}{4}\right] = \frac{n^{2}+3}{4}\) if \(n\) is odd.

Step 1: Identify Important Definitions

Before diving into the proof, let's first understand some terms and definitions that will be important for our proof. 1. Odd integers: An integer n is odd if there exists another integer k such that n = 2k + 1. 2. The floor function: \(\left[\frac{n^{2}}{4}\right]\) denotes the largest integer less than or equal to \(\frac{n^{2}}{4}\).

Step 2: Express Odd Integer as 2k+1

We are given that n is an odd integer, so we can express it as \(n = 2k + 1\) where k is any integer.

Step 3: Work with the Left-hand Side of the Equation

Let's handle the left-hand side of the equation first. We have: \(\left[\frac{n^{2}}{4}\right] = \left[\frac{(2k+1)^{2}}{4}\right]\) Next, we will expand the numerator: \(\left[\frac{(2k+1)^{2}}{4}\right] = \left[\frac{4k^2 + 4k + 1}{4}\right]\)

Step 4: Simplify the Left-hand Side of the Equation

We can now factor out the common factor of 4 from the numerator and simplify the expression: \(\left[\frac{4k^2 + 4k + 1}{4}\right] = \left[\frac{4(k^2 + k) + 1}{4}\right]\) As we have a term with only integer numbers and one term that is a fraction, then: \(\left[\frac{4(k^2 + k) + 1}{4}\right] = \left[\frac{4(k^2 + k)}{4}\right] + \left[\frac{1}{4}\right] = k^2+k\)

Step 5: Work with the Right-hand Side of the Equation

Now let's handle the right-hand side of the equation. We have: \(\frac{n^{2}+3}{4} = \frac{(2k+1)^{2}+3}{4}\) Next, we will expand the numerator: \(\frac{(2k+1)^{2}+3}{4} = \frac{4k^2 + 4k + 1 + 3}{4}\)

Step 6: Simplify the Right-hand Side of the Equation

We can now factor out the common factor of 4 from the numerator and simplify the expression: \(\frac{4k^2 + 4k + 1 + 3}{4} = \frac{4(k^2 + k) + 4}{4} = k^2+k\)

Step 7: Conclude the Proof

Since we have shown that both left-hand side and right-hand side expressions are equal to \(k^2 + k\), we conclude the proof: \(\left[\frac{n^{2}}{4}\right] = \frac{n^{2}+3}{4}\) if n is odd.

Key Concepts

Odd Integers

In discrete mathematics, odd integers are a fundamental concept. An odd integer is any integer that cannot be evenly divided by 2. This means if you take any integer \( n \), it is called odd if there exists an integer \( k \) such that \( n = 2k + 1 \).
This formula arises because when you multiply any integer \( k \) by 2, you get an even number. By adding 1, you move to the next odd integer.
Examples of odd integers include:

• 1 (which can be expressed as \( 2 \times 0 + 1 \))
• 3 (which can be expressed as \( 2 \times 1 + 1 \))
• 5 (which can be expressed as \( 2 \times 2 + 1 \))

Knowing what makes a number odd is crucial for understanding how they interact with various mathematical operations, such as those required in proofs and complex calculations.

Floor Function

The floor function is an essential mathematical concept. It is often represented using square brackets: \( \left[ x \right] \). The floor function of a real number \( x \), \( \left[ x \right] \), is defined as the greatest integer less than or equal to \( x \).
In simpler terms, the floor function rounds a number down to the nearest whole number less than or equal to the original number. For example:

• \( \left[ 3.7 \right] = 3 \) (since 3 is the largest integer \( \leq 3.7 \))
• \( \left[ -2.1 \right] = -3 \) (since -3 is the largest integer \( \leq -2.1 \))

The floor function is particularly useful in proofs involving integers, as it allows for a precise way to handle fractional results, ensuring that operations involving division are approached with integer values.

Proof Techniques

Proof techniques are methods employed to establish the truth of mathematical statements. In the exercise above, we use algebraic manipulation to prove an equation.
The proof starts by acknowledging definitions and properties of the involved mathematical elements such as odd integers and floor functions. Then, a strategy is employed:

• Substitution: Replacing odd integers with their expression \( n = 2k + 1 \).
• Simplification: Expanding and simplifying both sides of the equation.
• Comparative Analysis: Demonstrating that both simplified forms (left-hand side and right-hand side) are identical.
  

Using these steps means that you thoroughly check each component of the equation. By logically deducing and validating each part, you ensure the integrity and truth of the proof. This approach is central to many mathematical and logical conclusions.