Problem 7 - Discrete Mathematics with Applications

Question

Prove each directly. The square of an even integer is even.

Step-by-Step Solution

Answer

The square of an even integer, represented as \((2k)^2\), simplifies to \(4k^2\), which can be written as \(2(2k^2)\). Since the result is a multiple of 2, it is also an even integer. Therefore, the square of an even integer is even.

Step 1: Define an even integer

An even integer is any integer that can be written in the form 2k, where k is also an integer. Step 2: Square an even integer

Step 2: Square an even integer

To square an even integer, we will take the even integer in the form 2k and multiply it by itself. This can be represented as \((2k)^2\). Step 3: Simplify the squared even integer

Step 3: Simplify the squared even integer

Now, we will simplify and expand the expression \((2k)^2\). Using the distributive property, we get: \((2k)^2 = 4k^2\). Step 4: Determine if the result is even

Step 4: Determine if the result is even

We notice that the result, \(4k^2\), is a multiple of 2 since \(4 = 2 \cdot 2\). Therefore, we can represent the result as \(2(2k^2)\), which means the squared even integer is an even integer. Step 5: Conclusion

Step 5: Conclusion

The square of an even integer is also an even integer, as shown through the step-by-step direct proof.

Key Concepts

Even Integers

An even integer is a number that can be divided by 2 with no remainder. This simple characteristic defines them clearly from odd integers. Mathematically, any even integer can be expressed as \( 2k \), where \( k \) is some integer. This formula allows us to identify even numbers quickly and easily.

• If \( k = 0 \), then \( 2k = 0 \), which is even.
• If \( k = 1 \), then \( 2k = 2 \), which is even.
• If \( k = -1 \), then \( 2k = -2 \), which is also even.

Considering these steps helps us understand the flexibility and generality of even integers. It's important to remember that any integer can be transformed into an even integer by multiplying it by 2.

Direct Proof

The direct proof technique is a straightforward method for demonstrating mathematical statements. In this exercise, we used it to show that the square of an even integer is even. The approach involves directly applying known truths or definitions to arrive at the conclusion.
First, we start by taking a general even integer \( 2k \). Then, as per the requirements of the proof, it is squared using the expression \((2k)^2\). The calculation is done step-by-step:

• Multiply \( 2k \) by itself.
• Simplify the expression to \( 4k^2 \).

By proving each statement based on established mathematical properties, the direct proof becomes a powerful tool for verification.

Integer Properties

Integer properties are the characteristics that make numbers, like even and odd integers, behave in predictable ways. For this exercise, we particularly focus on the property of even numbers being multiples of 2. We begin with an even integer \( 2k \) and explore its squaring.
Here's what happens:

• The square of \( 2k \) becomes \((2k)^2\).
• This simplifies to \( 4k^2 \).

The outcome, \( 4k^2 \), is also even because 4 is a multiple of 2 (specifically, twice 2, as \( 4 = 2 \times 2 \)). Therefore, understanding how integers multiply and transform reinforces our focus on how consistent and reliable integer properties are in proofs.

Mathematical Proof

Mathematical proof is the process of establishing truth using logical reasoning and established axioms. In this case, we demonstrated that the square of an even integer is also even. Proofs are the backbone of mathematics and offer certainty.
To create a mathematical proof, you:

• Define the initial assumptions (an integer represented as \( 2k \)).
• Apply logical reasoning to modify or explore these assumptions (square \( 2k \)).
• Arrive at a conclusion (the result \( 4k^2 \) is even).

This systematic approach ensures that each statement logically follows from the previous one, solidifying the proof's validity and offering a clear, indisputable conclusion.