Problem 6 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The sum of any two odd integers a and b can be represented as a = 2n + 1 and b = 2m + 1, where n and m are integers. Adding these two odd integers gives a + b = (2n + 1) + (2m + 1), which simplifies to a + b = 2n + 2m + 2. Factoring out a 2 from the expression, we get a + b = 2(n + m + 1), which is the definition of an even integer as it is a multiple of 2. Therefore, the sum of any two odd integers is even.

Step 1: Define odd and even integers

An even integer is any integer that is exactly divisible by 2, meaning that it can be written in the form 2n, where n is an integer. An odd integer is any integer that cannot be exactly divided by 2, meaning that it can be written in the form 2n + 1, where n is an integer.

Step 2: Write the general form of two odd integers

Let's represent any two odd integers as a and b. Since they are both odd numbers, they can each be written in the form: a = 2n + 1 and b = 2m + 1, where n and m are integers.

Step 3: Sum the two odd integers

To prove that their sum is even, we add the two given odd numbers: a + b = (2n + 1) + (2m + 1)

Step 4: Simplify the expression

Next, we simplify the resulting expression by combining like terms: a + b = 2n + 2m + 2

Step 5: Factor out 2 from the expression

Now, we can factor out a 2 from the entire expression: a + b = 2(n + m + 1)

Step 6: Confirm that the result is even

Since the product of integer (n + m + 1) and 2, our resulting expression a+b = 2(n + m + 1) represents an even integer, as it is a multiple of 2.

Step 7: Conclusion

Thus, the sum of any two odd integers is an even integer, as proven by our step-by-step analysis above.

Key Concepts

Integer Representation

In mathematics, understanding integer representation is crucial when classifying numbers as even or odd. An integer is a whole number that can either be positive, negative, or zero. These numbers can further be divided into two categories based on their divisibility by 2: namely, even and odd numbers.

• Even Numbers: These are integers like -2, 0, 4, and 6, which when divided by 2, yield a remainder of zero. Mathematically, any even number can be expressed as: \( \text{even} = 2n \), where \( n \) is an integer.
• Odd Numbers: These integers, such as -3, 1, 5, and 9, have a remainder of one when divided by 2. The mathematical expression for an odd number is: \( \text{odd} = 2n + 1 \), where \( n \) is also an integer.

This classification aids in performing various operations, like addition and subtraction, without losing sight of these fundamental properties. Whether a number is even or odd has a significant impact on the results of mathematical computations. Understanding integer representation is foundational for performing higher-level mathematical proofs and reasoning.

Mathematical Proof

Mathematical proof is a logical process used to demonstrate the validity of a given statement. In our exercise, we are asked to prove that the sum of any two odd integers results in an even integer. The proof follows these clear steps:

• 1. Define the Parameters: Start by defining two odd integers, such as \( a = 2n + 1 \) and \( b = 2m + 1 \), where \( n \) and \( m \) are integers. This definition captures the essential property of odd numbers since each is one unit more than an even number.
• 2. Perform the Operation: Add these two odd integers: \( a + b = (2n + 1) + (2m + 1) \).
• 3. Simplify the Expression: By combining like terms, the expression simplifies to \( a + b = 2n + 2m + 2 \).
• 4. Factor to Prove Evenness: Factoring out 2, we get \( a + b = 2(n + m + 1) \), demonstrating that the sum is a multiple of 2, hence an even number.

The structured approach of a mathematical proof is all about ensuring each step logically follows from the last. This clarity is essential to show convincingly that the conclusion - the sum is even - is mathematically sound.

Addition of Integers

The addition of integers is a fundamental operation in mathematics and is crucial when dealing with concepts like even and odd numbers. Performing operations on integers involves understanding their intrinsic properties.

When you add two integers together, the result is another integer. Here's how the addition of even and odd integers works specifically:

• Even + Even: Adding two even numbers (like 2 + 4) results in an even number (6), since \( (2n + 2m) = 2(n + m) \).
• Odd + Odd: Adding two odd numbers always results in an even number because \( (2n + 1) + (2m + 1) = 2(n + m + 1) \), a crucial point we proved earlier in our mathematical proof section.
• Even + Odd: When an even number is added to an odd number, the result is always odd (like 2 + 3 = 5), expressed as \( 2n + (2m + 1) = 2(n + m) + 1 \).

These results hold due to how integers are defined and manipulated. By applying these simple rules, you can easily determine the parity (evenness or oddness) of the result when integers are added. The patterns they reveal are foundational, helping in both computational mathematics and theoretical proofs.