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
Best Answer
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.