2.10: Subsets

We learned that the “\(\in\)" symbol is used to indicate set membership: the element on the left is a member of the set on the right. A related but distinct notion is the idea of a subset. When we say \(X \subseteq Y\) (pronounced “\(X\) is a subset of \(Y\)"), it means that every member of \(X\) is also a member of \(Y\). The reverse is not necessarily true, of course, otherwise “\(\subseteq\)" would just mean “\(=\)". So if \(A\) = { Dad, Lizzy } and \(K\) = { Dad, Mom, Lizzy }, then we can say \(A \subseteq K\).

Be careful about the distinction between “\(\in\)" and “\(\subseteq\)", which are often confused. With \(\in\), the thing on the left is an element, whereas with \(\subseteq\), the thing on the left is a set. This is further complicated by the fact that the element on the left-hand side of \(\in\) might well be a set.

Let’s give some examples. Suppose that \(Q\) is the set { 4, { 9, 4 }, 2 }. \(Q\) has three elements here, one of which is itself a set. Now suppose that we let \(P\) be the set { 4, 9 }. Question: is \(P \in Q\)? The answer is yes: the set { 4, 9 } (which is the same as the set { 9, 4 }, just written a different way) is in fact an element of the set \(Q\). Next question: is \(P \subseteq Q\)? The answer is no, \(P \not\subseteq Q\). If \(P\) were a subset of \(Q\), that would imply that every member of \(P\) (there are two of them: 9 and 4) is also an element of \(Q\), whereas in fact, only 4 is a member of \(Q\), not 9. Last question: if \(R\) is defined to be { 2, 4 }, is \(R \subseteq Q\)? The answer is yes, since both 2 and 4 are also members of \(Q\).

Notice that by the definition, every set is a subset of itself. Sometimes, though, it’s useful to talk about whether a set is really a subset of another, and you don’t want it to “count" if the two sets are actually equal. This is called a proper subset, and the symbol for it is \(\subset\). You can see the rationale for the choice of symbol, because “\(\subseteq\)" is kind of like “\(\leq\)" for numbers, and “\(\subset\)" is like “\(<\)".

Every set is a subset (not necessarily a proper one) of \(\Omega\), because our domain of discourse by definition contains everything that can come up in conversation. Somewhat less obviously, the empty set is a subset of every set. It’s weird to think that \(\varnothing \subseteq Q\) when \(Q\) has several things in it, but the definition does hold. “Every" member of \(\varnothing\) (there are none) is in fact also a member of \(Q\).

One note about reading this notation that I found confusing at first. Sometimes the expression “\(a \in X\)" is pronounced “\(a\) is an element of \(X\)," but other times it is read “\(a\), which is an element of \(X\)". This may seem like a subtle point, and I guess it is, but if you’re not ready for it it can be a extra stumbling block to understanding the math (which is the last thing we need). Take this hypothetical (but quite typical) excerpt from a mathematical proof:

“Suppose \(k \in \mathbb{N} < 10\dots\)"

If you read this as “Suppose \(k\) is a natural number is less than 10," it’s ungrammatical. It really should be understood as “Suppose \(k\) (which is a natural number) is less than 10." This is sometimes true of additional clauses as well. For instance, the phrase “Suppose \(k \in \mathbb{R} > 0\) is the x-coordinate of the first point" should be read “Suppose \(k\), which is a real number greater than zero, is the x-coordinate of the first point."

I’ll leave you with a statement about numbers worth pondering and understanding:

\[\varnothing \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \Omega.\]