Problem 5 - Discrete Mathematics with Applications

Question

Prove each directly. The sum of any two even integers is even.

Step-by-Step Solution

Answer

In summary, to prove that the sum of any two even integers is even, let the two even integers be represented as \(2m\) and \(2n\), where m and n are also integers. Then, add them: \[(2m) + (2n)\]. Factor out the common factor 2: \[2(m + n)\]. The expression \(2(m+n)\) represents an even integer, so the sum of any two even integers is even indeed.

Step 1: Define even integers

An integer is considered even if it can be expressed in the form \(2n\), where n is some integer. This means that even integers are multiples of 2. Let's represent our two even numbers as \(2m\) and \(2n\), where m and n are also integers.

Step 2: Add the two even integers

Now, we will add the two even integers represented as \(2m\) and \(2n\): \[(2m) + (2n)\]

Step 3: Factor out the common factor

We can simplify the expression by factoring out the common factor, which in this case is 2: \[2(m + n)\]

Step 4: Conclude the proof

The expression \(2(m+n)\) represents an even integer (the product of 2 and the sum of the two integers m and n). Since we have shown that the sum of two even integers can be expressed in this form, we can conclude that the sum of any two even integers is indeed even. Thus, the statement is proven to be true.

Key Concepts

Even Integers

Even integers are numbers which are divisible by 2 without leaving any remainder. In simpler terms, if you divide an even number by 2, you get a whole number, or integer.
Some examples of even integers include:

• -4
• 0
• 6
• 28

You might notice that these numbers, when divided by 2, don't leave any leftovers. This characteristic of even numbers can be mathematically expressed in the form of a formula. An integer is even if it can be written as \(2n\), where \(n\) is another integer.

This means any even number is essentially a multiple of 2. However, it's not just about memorizing the \(2n\) formula. Understanding it helps us easily identify even numbers and solve more complex problems regarding integers. If you think of the number line, every alternate number starting from zero is an even number. This pattern is consistent because of how multiplication works with the base of 2.
By recognizing and using this pattern, we can handle even integers with more confidence in algebraic problems.

Direct Proof

In mathematics, a direct proof is a straightforward way of proving a statement. The idea is to start with something that is known to be true and use logical steps to arrive at the statement you want to prove. The main advantage of a direct proof is its clarity and simplicity.To apply this kind of proof to our exercise, we know from the definition of even integers that they can be written as \(2n\) for some integer \(n\).
Using direct proof, we aim to demonstrate that if you add two even integers, the result remains even. Here's a simple breakdown of how that works:

• Assume two even integers \(2m\) and \(2n\).
• Add them together to get \((2m) + (2n)\).
• Factor out the 2 to show \(2(m + n)\).

This step-by-step approach ensures everything remains grounded in logical progression.

Once you have framed the sum \(2(m + n)\), it is apparent that the total is still even since it is represented as the product of 2 and an integer \(m+n\). This conclusion supports the statement because the properties of an even number remain unchanged.

Integer Addition

Integer addition is the basic process of combining whole numbers to derive a sum. The procedure is simple, but its application plays a vital role in various mathematical proofs and problems, including the one we are discussing.When it comes to adding integers, the same basic rules apply as with any numbers:

• Add numbers directly if they have the same sign.
• If they have different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.

In our case, since we are dealing with even integers, they naturally simplify into forms of \(2n\).
When you add two even integers, such as \(2m\) and \(2n\), the operation becomes straightforward:

• Combine them to get \((2m) + (2n)\).
• Simplify to obtain \(2(m + n)\).

Brackets in an expression indicate that we are calculating the value inside first.
And by evaluating the expression \(m+n\), and multiplying the result by 2, we return to our even integer starting point.
Ultimately, integer addition helps us verify and understand why certain properties hold true, like the statement that the sum of any two even integers is even. By mastering these simple but crucial concepts, you'll have a solid foundation for future mathematical challenges.