Problem 54 - Discrete Mathematics with Applications

Question

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lceil\frac{n}{2}\right\rceil=\frac{n+1}{2}\) if \(n\) is odd

Step-by-Step Solution

Answer

We can prove that \(\left\lceil\frac{n}{2}\right\rceil=\frac{n+1}{2}\) if \(n\) is odd by the following steps: 1. Understand that an odd integer \(n\) can be written in the form \(n=2k+1\), where \(k\) is an integer. 2. Substitute \(n=2k+1\) into the equation: \(\left\lceil\frac{2k+1}{2}\right\rceil=\frac{2k+1+1}{2}\). 3. Simplify the right side of the equation to get: \(\frac{2(k+1)}{2}=k+1\). 4. Evaluate the left side to find that \(\left\lceil\frac{2k+1}{2}\right\rceil=k+1\) because \(\frac{2k+1}{2} = k+\frac{1}{2}\). 5. Compare both sides to see that they are equal, hence proving the given statement.

Step 1: Understanding Odd Integers

An integer \(n\) is odd if it can be written in the form \(n=2k+1\) where \(k\) is an integer. In other words, an odd integer is always one more than an even integer.

Step 2: Understanding the Ceiling Function

The ceiling function \(\left\lceil x \right\rceil\) is the smallest integer greater than or equal to \(x\). For example, \(\left\lceil 3.2 \right\rceil = 4\), and \(\left\lceil -1.7 \right\rceil = -1\).

Step 3: Substituting an Odd Integer for \(n\)

Since we need to prove the given equation for an odd integer, let's substitute \(n=2k+1\) into the equation, where \(k\) is an integer: \(\left\lceil\frac{2k+1}{2}\right\rceil=\frac{2k+1+1}{2}\)

Step 4: Simplifying the Right Side of the Equation

We can simplify the right side of the equation by combining like terms: \(\frac{2k+1+1}{2}=\frac{2k+2}{2}\) Next, we can factor out a common factor of \(2\): \(\frac{2(k+1)}{2}=k+1\)

Step 5: Evaluating the Left Side of the Equation

Now, let's evaluate the left side of the equation. To find the smallest integer greater than or equal to \(\frac{2k+1}{2}\), we can first observe that \(\frac{2k+1}{2} = k+\frac{1}{2}\). Since \(k\) is an integer, adding \(\frac{1}{2}\) to it gives a number exactly halfway between two consecutive integers. Thus, the smallest integer greater than or equal to \(k+\frac{1}{2}\) is simply \(k+1\).

Step 6: Comparing Both Sides and Concluding the Proof

We have now shown that \(\left\lceil\frac{2k+1}{2}\right\rceil=k+1\) and \(\frac{2k+1+1}{2}=k+1\) when \(n\) is an odd integer. Therefore, we have proven that \(\left\lceil\frac{n}{2}\right\rceil=\frac{n+1}{2}\) if \(n\) is odd.

Key Concepts

Understanding Odd Integers

When diving into the realm of integers in mathematics, we encounter various categories - one of these being 'odd integers'. An integer, by definition, is any number without a fractional component, and it can be positive, negative, or zero.

Odd integers, however, have a unique property: they are not divisible by two. More formally, we can define an odd integer as any integer that can be expressed in the form of

\( n = 2k + 1 \)

where \( k \) is also an integer. For example, the numbers 1, 3, and -11 are odd because they can be respectively written as \(2 \times 0 + 1\), \(2 \times 1 + 1\), and \(2 \times -6 + 1\). An important aspect to remember about odd integers is their behavior in operations: when you add or multiply two odd numbers, the result follows specific integer properties, a concept we'll explore later.

Demystifying The Ceiling Function

One of the intriguing concepts in discrete mathematics is the 'ceiling function', often symbolized by \( \left\lceil x \right\rceil \). It might sound complex at first, but it's quite a simple construct once we get to know it.

The ceiling function is employed to describe the smallest integer that is greater than or equal to a given real number, \( x \). It's like rounding up to the nearest whole number. For instance, \( \left\lceil 4.3 \right\rceil = 5 \) and \( \left\lceil -2.5 \right\rceil = -2 \). A good way to visualize this function is to think of a number line where you look for the integer that comes right after the specified number.

The ceiling function becomes particularly interesting when dealing with fractions or negative numbers, as it provides a consistent method to find an integer that adheres to its properties, irrespective of whether the input number is positive, negative, or even a fraction.

Integer Properties Worth Noting

In mathematics, dissecting the properties of integers is key to unlocking the mysteries of various algebraic problems. Integers are the building blocks of arithmetic, and their properties govern the rules for computation.

Some of the fundamental properties we must highlight are closure, commutativity, associativity, and distributivity. For example, the sum and product of any two integers is always an integer (closure property). This means that the integers are closed under addition and multiplication.

In addition, the commutative property indicates that the order in which we add or multiply integers does not affect the result, whereas associativity allows us to group numbers differently without changing the outcome. The distributive property elegantly bridges multiplication and addition by stating that \( a \times (b + c) = ab + ac \).

Understanding these integer properties enriches our mathematical toolbox, allowing us to manipulate and prove various assertions within the dizzying universe of numbers with confidence and clarity.