Comparing sets

Two sets can stand in several distinct relations to each other:

1. subset
2. superset
3. identity
4. proper subset
5. proper superset
6. disjoint
7. incomparable

Subset and superset

Given two sets \(A\) and \(B\), \(A\) is a subset of \(B\) iff every element of \(A\) is also an element of \(B\). In this case, one writes \(A \subseteq B\). For example, \(\left \{ a,b \right \} \subseteq \left \{ a,b,c,d \right \}\). Alternatively, one also says in this case that \(B\) is a superset of \(A\) (written \(B \supseteq A\)).

By the definition of subset, every set \(S\) is a subset of itself. The reasoning is simple. If \(S \subseteq S\), then every member of \(S\) must be a member of \(S\), which is obviously true (how could it be otherwise?).

In addition, the empty set is a subset of every set, including itself. This is because the empty set contains no elements at all, so it trivially holds that every member of the empty set is a member of every set.

Identity

Two sets are identical iff each one is a subset of the other. In formal terms, \(A = B\) iff both \(A \subseteq B\) and \(B \subseteq A\) hold. The reason for this is again simple:

1. If two sets \(A\) and \(B\) are identical, then they must contain exactly the same elements. But then every member of \(A\) is a member of \(B\), which implies \(A \subseteq B\). And it’s also the case that every member of \(B\) is a member of \(A\), so that we have \(B \subseteq A\), too.

2. In the other direction, if \(A \subseteq B\) and \(B \subseteq A\), then every member of \(A\) is a member of \(B\), and every member of \(B\) is a member of \(A\). But that can only happen if the sets are identical.

Proper subset and superset

We call \(A\) a proper subset of \(B\) (\(A \subsetneq B\)) iff \(A\) is a subset of \(B\) but \(A\) and \(B\) are not identical. In other words, every element of \(A\) is a member of \(B\), but not every element of \(B\) is a member of \(A\). We also say that \(B\) is a proper superset of \(A\) (\(B \supsetneq A\)).

Remember that it is possible for both \(A \subseteq B\) and \(B \subseteq A\) to be true — in this case, \(A = B\). But there can be no \(A\) and \(B\) such that \(A \subsetneq B\) and \(B \subsetneq A\).

Disjoint and incomparable sets

If there are two sets \(A\) and \(B\) such that neither \(A \subseteq B\) nor \(B \subseteq A\), then there can be only two scenarios. One option is that \(A\) and \(B\) are disjoint, which means that there is no \(x\) such that both \(x \in A\) and \(x \in B\) — the two sets have absolutely no overlap. In mathematical terms, \(A \cap B = \emptyset\). Alternatively, \(A\) and \(B\) might be incomparable. In this case the two sets have a limited overlap such that there is at least one \(x\) with both \(x \in A\) and \(x \in B\), but there are also \(a \in A\) and \(b \in B\) such that \(a \notin B\) and \(b \notin A\).

While the concept of two disjoint sets is simple enough, it becomes slightly counterintuitive when we consider the empty set. By definition, the empty set is a subset of every set, including itself. But by the definition of intersection, it also holds that \(A \cap \emptyset = \emptyset\) for any set \(A\). So this means that the empty set is a subset of every set \(A\) but also disjoint from it. In fact, \(A\) could even be the empty set itself, and thus the empty set is disjoint from itself! This is admittedly very unintuitive, and one could amend the definition of disjoint sets above to avoid all of this. But this edge case is harmless enough that it doesn’t really merit moving to a more complicated definition.

Remarks on notation

Similarity to \(\leq\) and \(\geq\)

Students sometimes confuse the symbols \(\subseteq\) and \(\supseteq\). To avoid that, just keep in mind that these symbols are modeled after \(\leq\) and \(\geq\) for numbers. Just like \(x \leq y\) means that \(x\) is at most as large as \(y\), \(x \subseteq y\) tells us that \(x\) contains at most all the elements of \(y\), and nothing else.

A note on \(\subset\)

You may occasionally come across the symbol \(\subset\) in other math texts. Some authors use \(\subset\) instead of \(\subseteq\), while others use it for \(\subsetneq\). As you might imagine, this is a frequent source of confusion. It is best to avoid \(\subset\) and use \(\subseteq\) and \(\subsetneq\) instead.

And then there’s \(\not\subseteq\)

Sometimes we might just want to say that \(A\) is not a subset of \(B\). We could paraphrase this, as in “it is not the case that \(A \subseteq B\)”. But mathematicians like to use symbols for common phrases, so there’s a dedicated symbol for this: \(\not\subseteq\). Careful, do not confuse \(\not\subseteq\) with \(\subsetneq\).

Here’s an overview of all the relevant notation:

Formula	means…
\(A \subseteq B\)	\(A\) is a subset of \(B\) (holds even if \(A = B\))
\(A \subsetneq B\)	\(A\) is a proper subset of \(B\) (\(A \subseteq B\) and \(A \neq B\))
\(A \not\subseteq B\)	\(A\) is not a subset of \(B\) (\(A \ni a \notin B\) for some \(a\))

As you might have expected, there’s corresponding counterparts for superset: \(\supseteq\), \(\supsetneq\), \(\not\supseteq\). There is no standardized symbol for sets being incomparable, although some authors like to use \(\sim\) for this purpose.

Recap