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Chapter 2: Problem 7

Prove that for all integers \(m\) and \(n,\) if \(m\) and \(n\) are even, then \(m+n\) is even.

Short Answer

Expert verified
The statement is true. If we assume \(m\) and \(n\) are even, it implies there exist integers \(j\) and \(k\) such that \(m = 2j\) and \(n = 2k\). Adding together these expressions gives \(m + n = 2(j + k)\), which is of the form \(2l\), where \(l = j + k\), proving that \(m + n\) is also even.

Step by step solution

01

Understanding What 'Even' Means

An integer is even if it can be written in the form \(2k\) where \(k\) is an integer. This means that every even number is a multiple of 2.
02

Assume \(m\) and \(n\) Are Even

Assume \(m\) and \(n\) are even integers. According to the definition of even numbers, there exist integers \(j\) and \(k\) such that \(m = 2j\) and \(n = 2k\), because each is a multiple of 2.
03

Adding \(m\) and \(n\)

We need to show that \(m + n\) is an even number, i.e., it can be written in the form \(2l\) where \(l\) is an integer. If we replace \(m\) and \(n\) with the expressions we found in the previous step, we get \(m + n = 2j + 2k = 2(j + k)\).
04

Conclude the Proof

From step 3, we see that \(m + n\) is in the form \(2l\), where \(l = j + k\), which is an integer. Thus, by definition, \(m + n\) is even.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Integers
Integers are the building blocks of mathematics, important to many types of mathematical proofs and operations, like addition and multiplication. Integers are whole numbers that can be either positive, negative, or zero.
They are expressed without any fractional or decimal component. Some examples of integers include -3, 0, 7, and 42. In the context of this exercise, our focus is specifically on even integers.
Even integers can be important in various mathematical settings because their properties allow mathematicians to make specific conclusions about different operations, as shown in this exercise.
Grasping the Idea of a Mathematical Proof
A mathematical proof is a logical argument that uses established facts, definitions, and mathematical reasoning to arrive at a statement being true. By using a series of defined steps, it allows us to prove ideas rigorously. The method depends on clearly stating assumptions, applying logic, and deriving conclusions.

In our exercise, the proof involves showing that the sum of two even integers is also even. A clear and methodical approach is used to show that if both numbers are multiples of 2, their addition results in another number that is a multiple of 2, thus being even. Proofs are crucial in mathematics as they provide certainty and understanding of statements across different areas.
Exploring the Addition of Even Numbers
To understand this concept, it's useful to remember the very definition of even numbers: they are multiples of 2. Given two even integers, we want to show their sum is also even.

Let's assume two numbers, say \(m\) and \(n\), and both are even, meaning they can be written as \(m = 2j\) and \(n = 2k\), where both \(j\) and \(k\) are integers.

When you add them, that's \(m + n = 2j + 2k\), which can be factored to \(2(j + k)\). Therefore, the expression \(j + k\) is an integer, signifying that the result is another multiple of 2, thus an even number.
  • This shows that adding two even numbers will always yield another even number.
  • It's a simple yet powerful demonstration of the consistent properties of integers under addition.
Through understanding this repetitive pattern, we gain insight into the regularity and reliability of mathematical operations.

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