Example 6.1.1.
A number is called prime if its only divisors are
This definition has some hidden parts: a more complete definition would be as follows.
Section 6.1 Definitions
Definitions are used in mathematics to label objects that have special properties, and to group all such objects together. Be careful with definitions: as stated by mathematicians, they often contain implicit conditions.
You should view a definition as a technical test or collection of technical tests that an object must pass before it can be given a specific label.
Worked Example 6.1.2.
Solution.
Let us test
- Yes,
- Yes,
- None of the numbers in the following list is an integer:
So
Now let us test
- Yes,
- Yes,
- However, clearly
So
Often, the first thing we do in mathematics is to look for ways to make testing our definition easier.
Proposition 6.1.3.
The proof is left to you as Exercise 6.12.1.
Worked Example 6.1.4.
Demonstrate that
Solution. (Sketch)
By the proposition, to check that
