Problem #20 in Chapter 5 uses an idea that we have *not* yet covered in class. It is no longer a required problem – you don’t have to do it – but I will give you extra credit if you turn in a solution. This is excellent practice in reading and applying a definition (just as we have been doing for the definitions of odd, even, divides, and so on). The problem relies on a new definition, that of congruence mod n – it appears in the book as Definition 5.1 on page 105, but I will also give it here:
Definition. Given integers a and b, and , we say “a is congruent to b mod n”, or , if
For example, if are told that , then we can conclude:
(by the definition of congruence), and
for some integer (by the definition of divides)
Hope this helps! Please write back and let me know if you have any questions.
Best of luck,
I wanted to give an example of writing a given set in set-theoretic notation – this should help out with some of the problems on your first homework assignment (Section 1.1).
Example: Write in set-theoretic notation:
In this case, you can see that the given set consists of all multiples of 5. A good way to approach problems like this is to start with one of the basic sets, for example
- the natural numbers
- the integers
In this example, I can see that multiplying every natural number by 5 should give me the set that I want. Therefore, I will use the formula , and the condition . Combining these in set-theoretic notation gives the solution:
Here are two ways to read this solution aloud:
- “the set of all such that is a natural number”, or
- “for each in the natural numbers, multiply by and include the result in the set”
I hope this helps – feel free to respond here if you any questions.