Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

Read the following proof that the sum of two odd numbers is even. Discuss the rules of inference used. Proof: Let \(x\) and \(y\) be odd numbers. Then \(x=2 k+1\) and \(y=2 j+1\) for some integers \(j\) and \(k\). By algebra, $$ x+y=2 k+1+2 j+1=2(k+j+1) $$ Note that \(k+j+1\) is an integer because \(k\) and \(j\) are integers. Hence \(x+y\) is even. Q.E.D.

Step-by-Step Solution

Answer

The sum of two odd numbers is even because their sum can be expressed as 2 times an integer, which is always even.

Step 1: Define Odd Numbers

Let’s define two odd numbers, say x and y. By definition, an odd number can be expressed in the form 2k+1, where k is an integer. Therefore, let:\( x = 2k + 1 \) and \( y = 2j + 1 \)where j and k are some integers.

Step 2: Add the Two Odd Numbers

Add the expressions for x and y:\( x + y = (2k + 1) + (2j + 1) \)

Step 3: Simplify the Sum

Simplify the addition:\( x + y = 2k + 1 + 2j + 1 = 2k + 2j + 2 = 2(k + j + 1) \)

Step 4: Identify the Result as an Integer

Since k and j are integers, their sum (k + j + 1) is also an integer. Let's call this integer m:\( m = k + j + 1 \)

Step 5: Conclude the Sum is Even

Since x + y = 2m, and 2 times any integer (m) is always even, we have:\( x + y = 2m \)Therefore, the sum of two odd numbers is even.

Key Concepts

rules of inference

In mathematics, rules of inference help us draw conclusions based on given premises. They are like the logical steps that guide our thought process to reach a conclusion. To understand the proof that the sum of two odd numbers is even, let's look at the key rules of inference applied:

• Definition of odd numbers: We start with the definition. Any odd number can be written as 2k + 1, where k is an integer. This is a premise that we accept as true.
• Addition of numbers: When we add two expressions of the form (2k + 1), we use the rule that allows us to combine like terms. This simplifies our expressions step-by-step.
• Simplification: We utilize algebraic manipulation (another concept explained below) to combine terms and simplify the expression further.
• Conclusion from arithmetic properties: Finally, since integers are closed under addition, the sum of integers is also an integer. Multiplying an integer by 2 will always yield an even number. Hence, we conclude the sum is even using basic properties of numbers.

This logical progression adheres to the rules of inference and ensures our conclusion is valid.

odd and even numbers

Understanding odd and even numbers is crucial to grasping the proof.

• Odd Numbers: An odd number is any integer not divisible by 2. Mathematically, you can express it as 2k + 1, where k is an integer. For example, 3 (2 * 1 + 1), 5 (2 * 2 + 1), and 7 (2 * 3 + 1) are all odd numbers.
• Even Numbers: An even number is any integer that is divisible by 2. You can write it as 2k, where k is an integer. Examples include 2 (2 * 1), 4 (2 * 2), and 6 (2 * 3).

In our proof, recognizing these definitions helps us see why combining a pair of odd numbers results in an even number. The simplification steps show how two odd numbers’ forms (2k + 1) combine to make an expression that fits the form of an even number (2m).

algebraic manipulation

Another vital concept is algebraic manipulation, which involves rearranging and simplifying expressions to reach a solution.

• Combining Like Terms: In our proof, we combine similar terms to make the equation simpler. For example:
  \[ x + y = 2k + 1 + 2j + 1 = 2(k + j + 1) \]shows how terms are grouped and simplified.
• Distribution: This rule allows us to distribute a number across terms within parentheses. This is crucial when we write 2(k + j + 1) to show the sum is essentially multiplied by 2.
• Arithmetic Properties: Understanding integer properties helps establish the conclusion. For example, recognizing that multiplying any integer by 2 will always yield an even number ensures the final step of the proof is logical and sound.

Mastering these algebraic techniques makes it easier to follow and construct mathematical proofs, ensuring your conclusions are valid and well-founded.