Demystifying Subset Symbols
A common question students face on exams is to determine whether one set, A, is a subset of another set, B. Similarly, a question might ask if set A is a strict subset of B. Questions like these often contain symbols that look similar, and one symbol that has an ambiguous meaning altogether. In this blog post, we work to demystify the symbols used for subsets and strict subsets, to help us better understand different textbooks and online resources on the topic.
Subsets
The following question will seem familiar to most discrete math students: given a set A and a set B, is A a subset of B?
When we talk about “subsets” (not “strict subsets”, which we’ll come back to), A is a subset of B if everything in A is also in B. For example:
If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subseteq B$.
(The $\subseteq$ symbol means “is a subset of”.)
That is to say:
$\{1, 2\} \subseteq \{1, 2, 3\}$,
because everything in the left set (1 and 2) is in the right set.
Note that every set is a subset of itself. For example:
$\{1, 2, 3\} \subseteq \{1, 2, 3\}$.
Strict Subsets
A similar but different question asks: is set A a strict subset of set B?
A “strict subset” (also called a “proper subset“) is different from a “subset”. Set A is a strict subset of set B if everything in A is in B; but, in addition to that, there’s something in B that isn’t in A.
Note the slightly different symbol for “is a strict subset of” in this example:
If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subsetneq B$.
(The $\subsetneq$ symbol means “is a strict subset of”.)
That is:
$\{1, 2\} \subsetneq \{1, 2, 3\}$.
However, {1, 2, 3} is not a strict subset of {1, 2, 3}. In fact, no set is a strict subset of itself.
Unambiguous Notation
When I write about subsets and strict subsets, I prefer to use completely unambiguous notation. To write that A is a subset of B, I write:
$A \subseteq B$.
And, to write that A is a strict subset of B, I write:
$A \subsetneq B$.
An Ambiguous Symbol
However, a third symbol exists that some authors use when discussing set theory:
$A \subset B$.
This symbol can be confusing for students who are searching through textbooks or online resources, because different authors use the symbol with no line underneath it to mean different things.
(1) Some authors use the $\subset$ symbol to mean “is a strict subset of”. That is, $A \subset B$ means that $A$ is a strict subset of $B$. These authors will then use $A \subseteq B$ to mean that $A$ is a subset of $B$.
(2) Other authors use the $\subset$ symbol to mean “is a subset of” (rather than strict subset). That is, $A \subset B$ means that $A$ is a subset of $B$. These authors will then use $A \subsetneq B$ to mean that $A$ is a strict subset of $B$.
(3) Finally, some authors (such as myself) prefer to avoid the $\subset$ symbol all together, using only $\subseteq$ and $\subsetneq$.
To visualize this ambiguity as a table:
\begin{array}{|l|c|c|c|}
\hline \\
& \color{red} \subset & \subseteq & \subsetneq \\
\hline \\
\text{Author 1} & \color{red} \text{Strict subset} & \text{Subset} & – \\
\hline \\
\text{Author 2} & \color{red} \text{Subset} & – & \text{Strict subset} \\
\hline \\
\text{Author 3} & – & \text{Subset} & \text{Strict subset} \\
\hline
\end{array}
What to Do as a Student
First and foremost, if the materials in your course use the $\subset$ symbol, be sure to check which meaning it has. Does your course material intend it to mean “is a strict subset of”, or to mean “is a subset of”? I recommend you check the front cover of your textbook or ask your professor.
But also, be careful about how subset and strict subset symbols are used when you’re searching for online resources or supplemental textbooks.
Any additional resources you find, beyond your course materials, may use the $\subset$ symbol to mean something different than it does in your lectures or readings. Be mindful about which meaning it has in any resources you find.
Typically, if you see the $\subseteq$ symbol used alongside the $\subset$ symbol, you’re dealing with something written by Author 1 in the table. That is, the author is likely using $\subset$ to mean “is a strict subset of”.
On the other hand, if you see the $\subsetneq$ symbol used alongside the $\subset$ symbol, you’re typically dealing with something written by Author 2 in the table. This author is likely using $\subset$ to mean “is a subset of”.
Two of the three symbols, though, are unambiguous.
Despite any confusion the symbol $\subset$ may cause, $\subseteq \textbf{always}$ means “is a subset of”, and $\subsetneq \textbf{always}$ means “is a strict subset of”, regardless of whether the $\subset$ symbol is used by an author. When you see either of these two symbols, you can be confident in their meaning.
A Quick Warning
While we didn’t discuss the meaning of $\not \subseteq$ in this blog post, I’d be remiss not to mention that it isn’t the same symbol as $\subsetneq$.
The $\not \subseteq$ symbol (which means “is not a subset of”) is different from the $\subsetneq$ symbol (which means “is a strict subset of”). For example, the following statement is correct: $\{1, 4\} \not \subseteq \{1, 2, 3\}$. However, the set $\{1, 4\}$ is $\textbf{not}$ a strict subset of $\{1, 2, 3\}$.
When writing $\subsetneq$, be careful that you don’t mistakenly write $\not \subseteq$.
Summary
Subset and strict subset notation can be inconsistent. For some authors, the $\subset$ symbol has one meaning (“is a strict subset of”), and for other authors it has a different meaning (“is a subset of”). If your course materials use the $\subset$ symbol, be sure you know which meaning is intended.
If you find additional materials online that use the $\subset$ symbol, look to see if the $\subseteq$ or $\subsetneq$ symbols are also used there, to provide context for the meaning of the $\subset$ symbol in that document.
The trick to demystifying the $\subset$ symbol is to remember that, regardless of who authored the material, the $\subseteq$ symbol always means “is a subset of”, and the $\subsetneq$ symbol always means “is a strict subset of”. The context those symbols provide can help you determine the meaning of $\subset$ when you see it.
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