Is it true that 2 = {2} ? (Where {2} is the set containing just 2)
If not why is this the case?
Furthermore can we say that 2 E {{2}}?
Thanks
$\displaystyle 2\ne\{2\}~$. $\displaystyle ~2$ is a natural number and $\displaystyle \{2\}$ is a set containing the number $\displaystyle 2$.
$\displaystyle 2\notin\{\{2\}\}$
Some axiomatic systems do define:
$\displaystyle \\0=\emptyset\\1=\{0\}\\2=\{0,1\}\\3=\{0,1,2\}\\ \vdots$ See Halmos.
Hello, kinhew93!
Are you really this confused on basic Set theory?
Is it true that 2 = {2}? .(Where {2} is the set containing just 2) .No
If not, why is this the case?
We have $\displaystyle 2$.
This is like having a banana.
We have $\displaystyle \{2\}$.
This is a paper bag containing a banana.
These are not the same.
[Furthermore can we say that: .$\displaystyle 2 \in \{\{2\}\}\,?$ .No
$\displaystyle 2$ is a banana.
$\displaystyle \{2\}$ is a bag containing a banana.
$\displaystyle \{\{2\}\}$ is a bag containing a bag containing a banana.
We can say: .$\displaystyle 2 \in \{2\}\,\text{ and }\,\{2\} \,\in\,\{\{2\}\}$
. . . and that's all.
Beware of using the word "group" when you really mean "set" -- the word "group" has a specialised meaning in mathematics which does not mean "set".
You are advised to use the word "set" whenever you mean "set" (although "collection" is okay-ish), as the word "set" has an equally precise meaning in mathematics.
The quickest way to answer your question is to say: no, $x \ne \{x\}$.
As for understanding why this is not the case, the best you can do is to go to the definition of a set and think about it.
Please do not try and believe that they are the same, just because you think they ought to be the same. As in many fundamental and subtle points in mathematics, your intuition may let you down.
Yes a banana is a banana and 2 is 2.How is it not the question I asked? I asked for clarification of the definition. Clearly in the real world a banana is a banana whether or not you say that it is part of a 'set'.
But 2 is not a set any more than a banana is a paper bag with or without bananas inside.
I like the paper bag and bananas analogy.
It may be of interest to know that we can build up 2 from nothing at all.
At set may have memebers (elements if you like)
Or it may have no members, in which case we call it the empty set and denote it thus {}.
One we have a paper bag, sorry set, for containing things we can put things in it.
I choose to put copies of the empty set thus.
{} the set with nothing in it.
{ {} } the set with one copy of the empty set in it AKA 1
{ {} {} } the set with two copies of the empty set in it AKA 2
{ {} {} {} } the set with three copies of the empty set in it AHA the number 3
This is how we can build up the natural counting numbers from nothing at all as sets in set theory.
Oh please, no.
$0$ is the empty set $\{\}$.
$1$ is a set containing the empty set: $1 = \{0\} = \{\{\}\}$
$2$ is a set containing both $0$ and $1$: $2 = \{0, 1\} = \{\{\}, \{\{\}\}\}$
$3$ is a set containing $0$, $1$ and $2$: $3 = \{0, 1, 2\} = \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}$
And so on.
What you wrote:
$\{\{\},\{\}, \{\}\}$
is not really a set of three things because a set can not contain more than one instance of the same thing.
Why not?
What about the set of roots of the equation (x-1)(x-1)(x-1)=0 ?
Edit
A set can contain whatever I want to put in it.because a set can not contain more than one instance of the same thing.
With regard to my equation example, the set of distinct roots of that equation is not the same as the set of all roots of that equation, nor does it possess the same number of members.
It is true that for some sets every member is defined to be different so that further structure can be applied, but that is not a requirement of sets.
If you want to put more than one instance of the same thing into a set, then it is called a bag. A set and a bag are not the same thing.
It is of absolute fundamental importance to the understanding of sets that they don't contain more than one copy of anything.
If you are a member of a club, then that club contains only one you, no matter how many membership cards you have.
$\{1, 2, 2, 3, 3, 4\} = \{1, 2, 3, 4\}$