Problem 3 - Discrete Mathematics with Applications

Question

Determine if each implication is trivially true. If \(n\) is a prime number, then \(n^{2}+n\) is an even integer.

Step-by-Step Solution

Answer

The given statement "If \(n\) is a prime number, then \(n^{2}+n\) is an even integer" is trivially true. This is because for all prime numbers, including the smallest prime number 2, the expression \(n^2+n\) results in an even integer.

Step 1: Understand the given statement and write down the hypothesis and conclusion

The statement given is: If \(n\) is a prime number, then \(n^{2}+n\) is an even integer. The hypothesis (if-part) of the statement is that \(n\) is a prime number, and the conclusion (then-part) is that \(n^{2}+n\) is an even integer. We need to show that the conclusion follows from the hypothesis.

Step 2: Prove the statement for prime numbers greater than 2

Let's first consider prime numbers greater than 2. Since all prime numbers greater than 2 are odd, we can represent them as \(n = 2k+1\) for an integer \(k\). Now, let's evaluate \(n^2 + n\) for these numbers: \(n^{2}+n = (2k+1)^{2}+(2k+1)\) Expanding the square, we get: \(= 4k^2 + 4k + 1 + 2k + 1\) Now, factor out 2 from the terms containing k: \(= 2(2k^2+3k+1)\) Since all of these terms are integers, the expression will result in an even integer. Therefore, for prime numbers greater than 2, the given statement holds.

Step 3: Check for the smallest prime number

The smallest prime number is 2. Now, let's evaluate \(n^2 + n\) for \(n=2\): \(n^2 + n = 2^2 + 2\) \(= 4 + 2\) \(= 6\) As 6 is an even integer, the given statement holds for n=2 as well.

Step 4: Conclusion

We have shown that for all prime numbers (including the smallest prime number 2), the given statement "If \(n\) is a prime number, then \(n^{2}+n\) is an even integer" holds. Therefore, the implication is trivially true.

Key Concepts

Prime Numbers

Prime numbers are the building blocks of the natural numbers. They are defined as numbers greater than 1 that have no divisors other than 1 and themselves. This means that a prime number cannot be formed by multiplying two smaller natural numbers together. For instance, 2, 3, 5, 7, 11, and 13 are all examples of prime numbers because they cannot be divided evenly by any number other than 1 and the number itself.

Understanding prime numbers is crucial for various areas of mathematics, including number theory and cryptology. In the exercise, we utilize the property of primes (specifically that all primes except 2 are odd) to demonstrate a specific mathematical statement—showing how prime numbers can relate to other concepts, such as even integers. The concept of prime numbers is deep-rooted in understanding mathematical structures and proving various theoretical propositions.

Even Integers

Even integers are the set of numbers that can be exactly divided by 2 without leaving any remainder. Mathematically, an even number can be defined as any integer of the form 2k, where k is also an integer. For example, numbers such as 2, 4, 6, and 8 are even integers because they can be written as 2 times 1, 2 times 2, 2 times 3, and 2 times 4, respectively.

In the context of the exercise, we are tasked with demonstrating that a certain expression involving a prime number results in an even integer. Understanding the definition and characteristics of even integers helps to simplify and solve the problem by showing that the expression, when simplified, conforms to the format of an even integer, which is pivotal in reaching the mathematical proof.

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a given statement. Proofs are integral to mathematics because they provide a means to establish certainty about mathematical claims. Generally, a proof is constructed by starting with known facts or axioms and applying logical reasoning to arrive at the conclusion.

In our exercise, the proof is structured by first establishing that the hypothesis (a number being prime) has certain properties, such as any prime number greater than 2 being odd. Then we apply algebra to show that the conclusion (the expression results in an even integer) logically follows. The steps in the solution exemplify the structured approach necessary for proving statements in discrete mathematics. By following this disciplined approach, we ensure that our conclusions are not only persuasive but also irrefutable within the framework of mathematical logic.