Set Builder Notation Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Set builder notation is an alternative to roster notation for describing sets. Set builder notation can describe some sets that roster notation can’t, but is correspondingly trickier to use correctly. This discussion gives you some initial practice reading and writing sets with set builder notation.

Reading Set Builder Notation

For each of the following sets, give an English description of the set, and a few examples of elements of the set. Where you can, try to give English descriptions that capture the intuition behind the set rather than just restating the predicate in English.

1. { x ∈ ℝ | x > 0 }
2. { 2n + 1 | n ∈ ℤ }
3. { n ∈ ℕ | n has no factors beside itself and 1 }
4. { x ∈ ℝ | x divided by 3 is an integer }
5. { 3n | n ∈ ℤ }

Writing Set Builder Notation

Can you describe the empty set using set builder notation? How many different ways of describing the empty set can you think of?

Use set builder notation to describe the following sets. There are generally multiple ways to describe a set with set builder notation, so it will be interesting to see how many different answers you come up with.

1. The set of real numbers between 1 and 6, inclusive
2. The set of integers that are multiples of 2 or of 5
3. The set of integer powers of 2 (i.e., the set of numbers equal to 2ⁿ for some integer n)
4. The set of perfect squares (i.e., integers that are the square of some other integer)
5. The set of real numbers that are the sine of some other real number

Can one use set builder notation to describe sets of anything besides numbers? If so, give some examples, or discuss how you could do it. If not, say why not.