Elementary Sets 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.)

This discussion gives you a chance to get used to the basic ideas of sets and their description using roster notation. Answer, comment on, ask questions about, etc. the following:

Use roster notation to describe the following sets, or explain why the set can’t be described using roster notation:

1. The set of all integer multiples of 3 between 0 and 9, inclusive.
2. The set of all integer multiples of 3 between 0 and 3000, inclusive.
3. The set of all non-negative integer multiples of 3.
4. The set of all integer multiples of 3.
5. The set of people who have posted to this discussion so far (include yourself, assuming you have or are about to post).
6. The set containing the sets {1,2}, {2,3}, and {3,4} (and nothing else).
7. The empty set.
8. The set of integers between 1 and 6, inclusive.
9. The set of real numbers between 1 and 6, inclusive.

Classify the following as true or false, and say why:

1. {a,b,c,a,d} is a set that contains 2 copies of the letter “a.”
2. {1,3,5} = {5,3,1}.
3. ∅ ⊆ {10,20}.
4. {10} = 10.
5. {10} = {{10}}.
6. 3 ∈ {1,3,5}.
7. 3 ⊆ {1,3,5}.
8. {10} ⊆ {{10}}.

The above questions spend a lot of time exploring the relationships between 10, {10}, and {{10}}. None of the suggested relationships actually hold (OK, I’ve given away a lot of “false” answers, but you still get to explain why they’re false). Give some correct descriptions of relationships between 10, {10}, and {{10}}, using set notations as appropriate.

Can you give a rigorous proof that the set of integer multiples of 6 between 0 and 18, inclusive, is a subset of the set of integer multiples of 3 between 0 and 18, inclusive?

What if you’re dealing with infinite instead of finite sets? For example, can you give a rigorous proof that the set of all integer multiples of 6 is a subset of the set of all integer multiples of 3?

And getting a little beyond sets, but illustrating an important point about mathematical thinking generally: to be really rigorous in the above proofs, we should probably be sure that everyone agrees on what an “integer multiple of 6” and “an integer multiple of 3” means. Can you give precise mathematical definitions of these phrases?