# A: Subsets, Elements, Set Logic; Cardinality versus Absolute Value Distinction?

I come from a Computer Science background, am new to this Stack Exchange and I am doing Discrete Math homework.

I am a little bit stumped on a question still, after shuffling through many notes and searching Math.stackexchange with tags (discrete-mathmatics) and (elementary-set-theory) to find a conclusion. I have a hunch of what is going on, and I suspect I might be overthinking things on what could be a simple trick question, so I'd immensely appreciate some help!

For a group of problems in the assignment, the Objective is simply to:

Give examples of two different elements in each set. Note that the elements are themselves sets.

The specific exercises in question, appear like this:

(?/a) T = {S ⊆ ℝ | |S| = 3}

(?/b) M = {S ⊆ ℤ | n ∈ S ⇒ |n| = 3}

Question a was easy. I just had to provide two sets with 3 elements each.

Question b is where I tripped down a rabbit hole. Yes, in this context, |S| is a cardinality function, because S denotes a set. What I am wondering then, is if |n| in the next question is simply an absolute value function (so in this case, only {3} , {-3} could be elements of M). Given the context, I am concerned that |n| might have something to do with cardinality that I am not understanding. I assume that the element n is not guaranteed to be a set, given that its lower case and conventionally used as a variable with numerical value in Discrete Mathamatics. This is why I suspect that absolute value might come into play when this |x| notation isn't being used on a Set. After all, I haven't seen anything about cardinality relating to just elements anywhere. Plus, I wonder if one intended to create an inequality condition with set logic; Isn't this how it would look?

Can someone please verify if this distinction is true? Greatly appreciated!

Yes, ordinarily (and I think in this case as well) when $n$ denotes a number, $|n|$ will denote it's absolute value. You're quite right to find that confusing in this context!
Moreover, in the context of (axiomatic) set theory, numbers themselves are defined as sets. For example $2$ can be defined as the set $\{0,1\}$ (where $1$ is the set $\{0\}$ and $0$ is just the empty set). In this case one could understand $|2|$ to be the cardinality of this set, that is, $2$. But I've never seen a situation where this would be the intended meaning (and anyway, the end result is the same whether you take the absolute value of $2$, or this kind of cardinality).
• However since the question is about members of $\mathbb Z$, and the members of $\mathbb Z$ are canonically defined as equivalence classes of pairs of natural numbers, the cardinality of any integer is $\aleph_0$. Which of course only confirms that here the absolute value is meant, as otherwise the condition could not be met to begin with. – celtschk Oct 30 '16 at 19:30