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

Question

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 $X$)

Answer 1

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. :)

Answer 2

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.

Answer 3

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.

Answer 4

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 $S$.

Answer 5

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:

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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, $$ (A\subseteq B)\leftrightarrow (\forall x)(x\in A\to x\in B)\leftrightarrow (\forall x\in A)(x\in B). $$