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Homework Chp 5 Update – problem #20

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Hi everyone,

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 n \in \mathbb{N}, we say “a is congruent to b mod n”, or a \equiv b \pmod n, if
n | (b-a).

For example, if are told that x \equiv 7 \pmod 3, then we can conclude:
3|(7-x)  (by the definition of congruence), and
7-x = 3k for some integer k (by the definition of divides)

Hope this helps!  Please write back and let me know if you have any questions.

Best of luck,
Prof. Reitz

Homework Hints: Set-theoretic notation

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Hi everyone,

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:  \{5,10,15,20,\ldots \}

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 \mathbb{N} = \{1,2,3,4, \ldots \}
  • the integers \mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}

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 5n , and the condition n\in \mathbb{N}.  Combining these in set-theoretic notation gives the solution:

Solution \{ 5n : n\in \mathbb{N} \}

Here are two ways to read this solution aloud:

  • “the set of all 5n such that n is a natural number”, or
  • “for each n in the natural numbers, multiply n by 5 and include the result in the set”

I hope this helps – feel free to respond here if you any questions.

Best,
Prof. Reitz