# The difference between ∈ and ⊂

I had a task where I had to figure if the argument was true or not.

$A=\{ n ∈ ℤ \mid n^2 < 5 \}, \quad B=\{ 7, 8, \{2\}, \{2, 7, 8\}, \{\{7\}\} \}$

The first one was $\{-1, 2\} ∈ A$ and the answer to this was not true since the set is not an integer.

The second one was $\{-1, 2\} ⊂ A$ and the answer to this was true since -1 and 2 are elements of $A$.

I just can't seem to understand the difference between ∈ and ⊂ in this task and why the first argument was not true. Also it confuses me that the answer to the first one was that not true because the set is not an integer but aren't -1 and 2 integers?

There is other parts of this task that also add more confusion:

{2, 7, 8} ∈ B Answer: True since {2, 7, 8} is an element of set B

{2, 7, 8} ⊂ B Answer: Not true, for example 2 is not an element in set B

• you seem stuck on notation, do you understand what these symbols mean ? – user451844 Oct 7 '17 at 18:18

Intuitively, a set is a bag of things.

One of the things in the bag is an element. So we could say $\color{limegreen}{\large{\bullet}}\in S$.

On the other hand, if you pick things from your bag and stick them in a new bag, you've got a subset. So we can say $\{\color{cyan}{\large{\bullet}},\color{limegreen}{\large{\bullet}},\color{red}{\large{\bullet}}\}\subset S$.

More concretely, we can take any integer from $\mathbb{Z}$ and say $1\in\mathbb{Z}$ or that $5\in\mathbb{Z}$. But when we take integers from $\mathbb{Z}$ and put them in another set, we would say $\{1,5\}\subset\mathbb{Z}$ or $\{3,4,1\}\subset\mathbb{Z}$ or even $\{\cdot\}\subset\mathbb{Z}$. In a nutshell, $\in$ is used for objects in the set but $\subset$ is used for collections of objects in the set.

Simply:

$x\in y$ if $x$ is an element of $y$. An example of this is $2\in\{a, 2, \pi\}.$

$x\subset y$ if every element of $x$ is an element of $y$. An example of this is $\{2, \pi\}\subset\{a, 2, \pi\}$

For your problem specifically:

1. The set of $\{-1, 2\}$ is not an element of the set of integers whose squares are less than $5$.

2. The set of $\{-1, 2\}$ is a subset of $A$ because all elements of $\{-1, 2\}$, both $-1$ and $2$, are integers whose squares are less than $5$.

First of all we use $\in$ for elements of a set and $⊂$ for subset of a set.

For the first part note that $\{1,2\}$ isn't an element of the set $A$, in fact it's a subset of it.

For the second part, note that $B$ also contains subsets (as elements), therefore $\{2,7,8\}$ is an element of $B$, but not a subset, as there is no element $2$ in $B$.

$\in$ stands for "belongs to".For eg. an element may belong to a set.

$\subset$ is the symbol for subset .For eg. one set can be a subset of another set if all elements of that set is included in the later set.

You must be careful on what you are applying $\in$ or $\subset$.

A set (e.g. $A$ or $B$) has individual elements. These elements are abstract objects (e.g., in $A$ they are integers), but sometimes confusingly these elements can be also sets ($B$ has elements that are integers as well as elements that are sets).

$\in$ means "is an [individual] element of." The elements of $A$ are integers $-2, -1, 0, 1, 2$, while the set $\{-1,2\}$ is not one of these integers. The set $\{2,7,8\}$ is one of the elements of $B$, so $\{2,7,8\} \in B$.

$\subset$ means "is a subset of." The left-hand side of $\subset$ should be a set whose elements are also in the set on the right-hand side of $\subset$. The set $\{-1,2\}$ has two elements $-1$ and $2$, both of which are in $A$, so we write $\{-1,2\} \subset A$. The set $\{2,7,8\}$ has three elements $2,7,8$, but not all of them (actually none of them) are elements of $B$, so $\{2,7,8\} \not \subset B$.