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 :)

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- Jan 14th 2014, 07:39 AMkinhew93Is it true that x = {x} ?
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 :) - Jan 14th 2014, 08:01 AMPlatoRe: Is it true that x = {x} ?
- Jan 14th 2014, 08:15 AMSorobanRe: Is it true that x = {x} ?
Hello, kinhew93!

Are you really this confused on basic Set theory?

Quote:

Is it true that 2 = {2}? .(Where {2} is the set containing just 2) .No

If not, why is this the case?

We have .

This is like having a banana.

We have .

This is a paper bag containing a banana.

These arethe same.*not*

Quote:

Furthermore can we say that: . .No

is a banana.

is a bag containing a banana.

is a bag containing a bag containing a banana.

We can say: .

. . . and that's all. - Jan 14th 2014, 10:42 AMkinhew93Re: Is it true that x = {x} ?
Thanks - i know its simple I was just checking exactly what the notation meant, eg is it like a bag containing a banana or just a 'group' of one banana. thanks for clarifying

- Jan 21st 2014, 05:30 AMHallsofIvyRe: Is it true that x = {x} ?
- Aug 1st 2015, 11:59 AMkinhew93Re: Is it true that x = {x} ?
- Aug 1st 2015, 12:17 PMMatt WestwoodRe: Is it true that x = {x} ?
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. - Aug 1st 2015, 12:20 PMMatt WestwoodRe: Is it true that x = {x} ?
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. - Aug 1st 2015, 12:20 PMstudiotRe: Is it true that x = {x} ?Quote:

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.

:) - Aug 1st 2015, 12:24 PMMatt WestwoodRe: Is it true that x = {x} ?
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. - Aug 1st 2015, 12:27 PMstudiotRe: Is it true that x = {x} ?
Why not?

What about the set of roots of the equation (x-1)(x-1)(x-1)=0 ?

Edit

Quote:

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. - Aug 1st 2015, 12:45 PMMatt WestwoodRe: Is it true that x = {x} ?
- Aug 1st 2015, 12:47 PMMatt WestwoodRe: Is it true that x = {x} ?
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.

$\{1, 2, 2, 3, 3, 4\} = \{1, 2, 3, 4\}$ - Aug 1st 2015, 12:48 PMstudiotRe: Is it true that x = {x} ?
A true Mexican Standoff ?

:) - Aug 1st 2015, 12:49 PMMatt WestwoodRe: Is it true that x = {x} ?