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Beginning Activity Beginning Activity 1: Definition of Even and Odd Integers
A4US

Definitions play a very important role in mathematics. A direct proof of a proposition in mathematics is often a demonstration that the proposition follows logically from certain definitions and previously proven propositions. A definition is an agreement that a particular word or phrase will stand for some object, property, or other concept that we expect to refer to often. In many elementary proofs, the answer to the question, ā€œHow do we prove a certain proposition?ā€, is often answered by means of a definition. For example, in Progress CheckĀ 1.2, all of the examples should have indicated that the following conditional statement is true:

If \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer.
In order to construct a mathematical proof of this conditional statement, we need a precise definition of what it means to say that an integer is an even integer and what it means to say that an integer is an odd integer.

Definition.

An integer \(a\) is an even integer provided that there exists an integer \(n\) such that \(a = 2n\text{.}\) An integer \(a\) is an odd integer provided there exists an integer \(n\) such that \(a = 2n + 1\text{.}\)

Using this definition, we can conclude that the integer 16 is an even integer since \(16 = 2 \cdot 8\) and 8 is an integer. By answering the following questions, you should obtain a better understanding of these definitions. These questions are not here just to have questions in the textbook. Constructing and answering such questions is a way in which many mathematicians will try to gain a better understanding of a definition.

1.

Use the definitions given above to

(a)

Explain why 28, \(-42\text{,}\) 24, and 0 are even integers.

(b)

Explain why 51, \(-11\text{,}\) 1, and \(-1\) are odd integers.

It is important to realize that mathematical definitions are not made randomly. In most cases, they are motivated by a mathematical concept that occurs frequently.

2.

Are the definitions of even integers and odd integers consistent with your previous ideas about even and odd integers?