What is the difference between the relations "$\in$" and "$\subseteq$" ? Don't they both mean that something is an element of a set? Are they interchangeable in some or all situations?


$x \in A$ ($X$ is an element of the set $A, X$ is in $A, A$ contains $X$)

$x \subseteq A$ ($X$ is an element of the set $A, X$ is in $A, A$ contains $X$)

  • $\begingroup$ see here $\endgroup$ – user 1 Apr 3 '15 at 17:25
  • 1
    $\begingroup$ one is about membership other about inclusion $\endgroup$ – user 1 Apr 3 '15 at 17:26
  • 3
    $\begingroup$ No, they do not both mean that something is an element of a set. $x\in A$ means that $x$ is an element of $A$; $x\subseteq A$ means that $x$ is a subset of $A$, which in turn means that every element of $x$ is an element of $A$. If $A$ is the set of all living persons, $\text{Bruno Schiavo}\in A$, but $\text{Bruno Schiavo}\nsubseteq A$: you cannot be a subset of $A$, since you are not a set in the first place. $\endgroup$ – Brian M. Scott Apr 3 '15 at 17:29
  • 2
    $\begingroup$ In this context the word contains can be ambiguous, as it may mean either contains as an element or contains as a subset, and these are very different concepts. It’s better to avoid the word altogether. $\endgroup$ – Brian M. Scott Apr 3 '15 at 17:31
  • 1
    $\begingroup$ Surely this must be a duplicate. $\endgroup$ – MJD Apr 3 '15 at 17:37

The following are true:

$$rock\in \{rock,paper,scissors\}$$ $$\{rock\} \subseteq\{rock,paper,scissors\}$$ $$rock\not\subseteq \{rock,paper,scissors\}$$ $$\{rock\}\not\in\{rock,paper,scissors\}$$

editted to make it clearer.

  • 4
    $\begingroup$ The third one is wrong if $x = \{y,z\}$ and the fourth one is wrong if $y = \{x\}$. $\endgroup$ – Trevor Wilson Apr 3 '15 at 18:08

Comment: I'm writing this answer because I find it very odd that no one has even mentioned the name for the symbol "$\subseteq$". This symbol means "subset." It may help to review some basic terminology before you can really understand avid19's answer.

Notation and terminology (what $\in$ and $\subseteq$ mean and a few more symbols):

If $A$ is a set and $x$ is an entity in $A$, we write $x\in A$ and say that $x$ is an element of $A$. If we write $x\not\in A$, then this means that $x$ is not an element of $A$.

Given two sets $A$ and $B$, it may be the case that all elements of $A$ are also elements of $B$. This may be written as $A\subseteq B$, and we say that $A$ is a subset of $B$. Also, we may write $B\supseteq A$ and say that $B$ is a superset of $A$. If $A$ is a subset of $B$, but there are elements of $B$ that are not in $A$, then we say that $A$ is a proper subset of $B$, and this is written as $A\subset B$.

Can you understand avid19's answer now?

It may be helpful to note that the following is more rigorous formulation of the notion of what it means for a set be a subset of another:

Formal definition of subset: Suppose $A$ and $B$ are sets. We say that $A$ is a subset of $B$, written $A\subseteq B$, provided that for all $x$, if $x\in A$, then $x\in B$. That is, more formally, $$ (A\subseteq B)\leftrightarrow (\forall x)(x\in A\to x\in B)\leftrightarrow (\forall x\in A)(x\in B). $$


There is a fundamental difference between $\in$ and $\subseteq$.

Let's say we have a set $S$, it contains some balls.

If we want to talk about a ball in the set $S$, we use $\in$, so $b\in S$ means that $b$ is one of the balls found in the set.

On the other hand, if we want to talk about a bag which only contains balls from the set $S$, we use $\subseteq$, and $Z\subseteq S$ means that $Z$ is a bag which only contains balls found in the set $S$.


If you use the $\in$ mark, that is only for one element.

If you use the $\subseteq$ mark, that is for a set.

Let us have $\mathbb{N}$ as example, in that case, $1\in\mathbb{N} $, but if you take $X$ as the set of odd numbers: $X\subseteq\mathbb{N}$.

Hope you can understand the difference, it is really simple. :)

  • $\begingroup$ Can we write X ∈ N? If so, where's the need for the symbol "⊆"? What if we don't know if a given element is a set or not? $\endgroup$ – Bruno Schiavo Apr 3 '15 at 17:29
  • 1
    $\begingroup$ You can't write $X \in \mathbb{N}$, because $X$ is a set with more than one(actually infinite) elements, in that case you will need $\subseteq$ $\endgroup$ – Atvin Apr 3 '15 at 17:29
  • $\begingroup$ Hope you understand the difference, someone above already posted all the possibilities. :) $\endgroup$ – Atvin Apr 3 '15 at 17:32
  • $\begingroup$ Sets can be elements of other sets, even in naive set theory. (And in axiomatic set theory, everything is a set.) So whether or not the object on the left is a set is not the essential difference between $\in$ and $\subseteq$. $\endgroup$ – Trevor Wilson Apr 3 '15 at 18:12
  • $\begingroup$ Also, I think your comment is misleading because it is wrong to say $Y \in \mathbb{N}$ when $Y = \{3\}$ (for example) even though this set $Y$ has only one element. $\endgroup$ – Trevor Wilson Apr 3 '15 at 18:13

As I said in comment $x\in A$ is about membership of the element $x$ in $A$. But $X \subseteq A$ is about inclusion of the subset $X$ of $A$ in $A$. see also here and here.

  • $\begingroup$ What's the difference between ⊂ and ⊆? $\endgroup$ – Bruno Schiavo Apr 3 '15 at 17:31
  • $\begingroup$ it depends on books: some use the first as strict, some use the first as second $\endgroup$ – user 1 Apr 3 '15 at 17:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.