Is {} same as {{}} set theory problem

**gsc** · October 13th 2007, 05:01 AM

Ok basic stuff if I have a set A={d,0,{},{{}},1} is this a 5 element set or is {} and {{}} the same ....I think it is 5 elements and thus has 32 elements in its power set but not too sure.....some maths brain put me outta my misery please!

**topsquark** · October 13th 2007, 05:22 AM

Originally Posted by gsc

Ok basic stuff if I have a set A={d,0,{},{{}},1} is this a 5 element set or is {} and {{}} the same ....I think it is 5 elements and thus has 32 elements in its power set but not too sure.....some maths brain put me outta my misery please!

Strictly speaking $\{ \} = \emptyset$ , so $\{ \{ \} \} = \{ \emptyset \}$ which is "the set of elements where the only element is the empty set." It is slightly different in concept to the empty set.

For example if you were working with sets of sets then $\{ \emptyset \}$ would have to be your "null" element, not simply $\emptyset$ .

-Dan

**Ben Tillman** · October 13th 2007, 05:22 AM

You are correct. The set {{}} contains a set (that it's the empty set is of no consequence), hence it's non-empty, as opposed to {}.

EDIT: Too slow ..

**Soroban** · October 13th 2007, 07:11 AM

Hello, gsc!

Here's my baby-talk approach to this . . .

Is {} and {{}} the same ?

Think of a set as a paper bag.

Then $\{3\}$ is a bag containing a 3.
. . and $\{p,q\}$ is a bag containing $p$ and $q$ .

And $\emptyset \:=\:\{\,\}$ is a bag containg nothing, an empty bag.

Note that: . $\{0\}$ is not empty; the bag contains a 0.
. . . . . $\{\text{nothing}\}$ is not empty; the bag contains a word.

Then: . $\{\emptyset\} \:=\:\{\{\:\}\}$ is not empty; the bag contains another bag.

**gsc** · October 13th 2007, 03:21 PM

It's all clear now!
Thanks for your replies much appreciated....gives me faith in human nature!!! (all you maths genius' helping me, a maths mongo!!). Topsquark thanks for the jokes too.

Set theory is great but I'm not surprised that Cantor ended up in the loony bin....it all does your head in!

**Jhevon** · October 13th 2007, 03:21 PM

Originally Posted by Soroban

Hello, gsc!

Here's my baby-talk approach to this . . .

Think of a set as a paper bag.

Then $\{3\}$ is a bag containing a 3.
. . and $\{p,q\}$ is a bag containing $p$ and $q$ .

And $\emptyset \:=\:\{\,\}$ is a bag containg nothing, an empty bag.

Note that: . $\{0\}$ is not empty; the bag contains a 0.
. . . . . $\{\text{nothing}\}$ is not empty; the bag contains a word.

Then: . $\{\emptyset\} \:=\:\{\{\:\}\}$ is not empty; the bag contains another bag.

i like that analogy for some reason

**Jhevon** · October 13th 2007, 03:25 PM

Originally Posted by gsc

Set theory is great but I'm not surprised that Cantor ended up in the loony bin....it all does your head in!

hehe. you always hear people say things like that..."math makes you crazy!" I don't think so. i think if a mathematician went crazy, he was inclined to being crazy to begin with and would have ended up crazy even if he had not gotten involved in math. maybe the thing is that math does not make you crazy, but it appeals to you if you are somewhat crazy to begin with

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Thank you all very much!

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