Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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

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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, “How do we prove a certain proposition?”, is often answered by means of a definition. For example, in Progress Check 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.

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.

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.