Homework Hints: Set-theoretic notation

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.

Prof. Reitz


2 thoughts on “Homework Hints: Set-theoretic notation

  1. Dear Prof. Reitz,

    Thank you for the examples. It helped me for the homework. But, I have confused about the question: {x is an element of “Z” : |2x| <5}. Can you please explain me little bit about this question?



    1. Hi Fuzail,

      This one looks a little strange, because we are expecting a formula to come first, and instead we have x\in \mathbb{Z}. This is something common in mathematics that doesn’t exactly follow the rule, in which we describe the type of number (integers) in the first part of the set-theoretic notation – really, it means the same thing as this: \{ x : x\in \mathbb{Z} and |2x|<5 \}.

      I hope this helps – feel free to write back if not.
      -Prof. Reitz

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