What is the difference between the relations $\in$ and $\subseteq$?

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?

Like:

$x \in A$ ($X$ 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 $X$)

• see here – user 1 Apr 3 '15 at 17:25
• one is about membership other about inclusion – user 1 Apr 3 '15 at 17:26
• 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. – Brian M. Scott Apr 3 '15 at 17:29
• 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. – Brian M. Scott Apr 3 '15 at 17:31
• Surely this must be a duplicate. – 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.

• The third one is wrong if $x = \{y,z\}$ and the fourth one is wrong if $y = \{x\}$. – 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. :)

• 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? – Bruno Schiavo Apr 3 '15 at 17:29
• 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 $\subseteq$ – Atvin Apr 3 '15 at 17:29
• Hope you understand the difference, someone above already posted all the possibilities. :) – Atvin Apr 3 '15 at 17:32
• 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 $\subseteq$. – Trevor Wilson Apr 3 '15 at 18:12
• 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. – 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.

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