-2

Introductory question about set and subsets. Found this paragraph is my textbook:

Let $A$ be a set and let $B = \{A, \{A\}\}$. Then, since $A$ and $\{A\}$ are elements of $B$, we have $A\in B$ and $\{A\} \in B$. It follows that $ \{A\}\subset B$ and $\{\{A\}\} \subset B$. However, it is not true that $A\subset B$.

I don't understand why $A$ is not a subset of $B$. As I understand it, all the elements of $A$ are elements of $B$... Is this a bit of semantics where it is not the elements of $A$ that are elements of $B$ but rather the set $A$ that is an element of $B$? If it is semantics is it even important to know?

Arctic Char
  • 16,007
YinGroen
  • 7
  • 1
    There is a difference between $A$ and ${ A }$. – Mauro ALLEGRANZA Aug 31 '21 at 08:10
  • Consider e.g. $A = { 1,2,3 }$. It has three elements: $1,2$ and $3$, while ${ A }$ has only one element. – Mauro ALLEGRANZA Aug 31 '21 at 08:11
  • It is not true that all elements of $A$ are elements of $B$... The elements of $B$ are only two: $A$ and ${ A }$ (they are the "objects" listed between ${ \ldots }$ in the definition of $B$) while we do not know the elements of $A$ itself: maybe none... – Mauro ALLEGRANZA Aug 31 '21 at 08:15
  • See also this post – Mauro ALLEGRANZA Aug 31 '21 at 08:29
  • If you don't understand something, the first step is to try it with some concrete objects. If $B$ has two elements, taking $A$ that has at least three will be an obvious giveaway. Actually, even if you do understand it, it's good to have a concrete example. – Asaf Karagila Aug 31 '21 at 09:07
  • You recieved 5 answers to your question. Is any of them what you needed? If so, consider accepting the best answer and upvoting all useful answers you got. That's how the site works. – 5xum Sep 01 '21 at 06:07

4 Answers4

2

No it is not true that all elements of $A$ are elements of $B$. Consider for example $A=\{1,2\}$. Then $B$ is the set $\{\{1,2\},\{\{1,2\}\}\}$. Note that $1$ and $2$ are not elements of $B$. Only a set containing both $1$ and $2$ is an element of $B$. This makes the set ($A$) an element of $B$, but not a subset.

spinosarus123
  • 1,885
1

As I understand it, all the elements of A are elements of B...

No. There are two elements of $B$. One element of $B$ is $A$, the other is $\{A\}$. None of the two elements is an element of $A$.

Take, for example, an example where $A=\{1,2,3\}$. Then, $B$ has two elements, one of them is $\{1,2,3\}$, the other is $\{\{1,2,3\}\}$.

On the other hand, $A$ has the element $1$, and $1$ is not an element of $B$, since $1$ is not equal to $\{1,2,3\}$, and it is not equal to $\{\{1,2,3\}\}$.

5xum
  • 123,496
  • 6
  • 128
  • 204
0

Suppose we assume that $A \subseteq B$,

then every element of $A$ should be in $B$. So take an element $x$ in $A$ which is in $B$. But $B$ has only two elements $A$ and $\{A\}$ so $x$ has to be one of them. It is clear that $\{A\}$ can not be an element of $A$. Also $x$ can not be $A$ because $A$ can not be an element of itself.

So we conclude that $A\subseteq B $ is not true.

Infinity_hunter
  • 5,369
  • 1
  • 10
  • 30
0

To add to the answers here, and hopefully cut to the heart of your confusion, sets are denoted by curly braces. $A$, while a set, is an element of set $B$, so it needs to go in curly braces. So $\{A\}$ not $A$. If in doubt, it may help to replace the elements with numbers. So $B=\{1, 2\}$ and $\{1\} \subseteq B$. Other answers have covered your other confusion

Ten O'Four
  • 1,056