Confusion over empty sets

The following question is very confusing to me.

I can't decide if the set

\{ \emptyset \}

has no elements or one element! Would anybody be able to point me in the right direction to working this out? I'm sure the answers to (iii) and (iv) will become clear once I understand (ii). Thank you in advance!

Confusion over empty sets.PNG5.3KB

Reply 1

atsruser

Original post by foorganders

The following question is very confusing to me.

I can't decide if the set

\{ \emptyset \}

Well, what is the difference between

A=\{\}

and

B=\{\emptyset\}

?

What kind of thing is the

\emptyset

that seems to be living in

B

Reply 2

foorganders

Original post by atsruser

Well, what is the difference between

A=\{\}

and

B=\{\emptyset\}

?

What kind of thing is the

\emptyset

that seems to be living in

B

So it has one element, even though that element is the empty set? And would (iii) and (iv) both have two elements?

Reply 3

ghostwalker

Original post by foorganders

So it has one element, even though that element is the empty set?

Yes.

And would (iii) and (iv) both have two elements?

(iii) clearly has two elements.

However, for (iv), multiplicity doesn't count, there is only one element there, even though it's been written twice.

Note, the following two sets are equal: {a} and {a,a,a,a,a,a}. Every element of the first set is in the second, and every element of the second set is in the first.

(edited 8 years ago)

Reply 4

foorganders

Original post by ghostwalker

Yes.

(iii) clearly has two elements.

However, for (iv), multiplicity doesn't count, there is only one element there, even though it's been written twice.

Note, the following two sets are equal {a} and {a,a,a,a,a,a}. Every element of the first set is in the second, and every element of the second set is in the first.

Ah ok, thank you for the clarification. So would the set

\{ \{ \emptyset \}, \{ \emptyset \}, \{ \emptyset \} \}

have three elements? Or one? And if it is one, then how would you go about constructing a similar set, but with three elements?

(edited 8 years ago)

Reply 5

ghostwalker

Original post by foorganders

Ah ok, thank you for the clarification. So would the set

\{ \{ \emptyset \}, \{ \emptyset \}, \{ \emptyset \} \}

have three elements? Or one?

There is only one distinct element there. So, one.

And if it is one, then how would you go about constructing a similar set, but with three elements?

Sorry, you'll have to clarify that. How can it be similar and have three elements?

Reply 6

foorganders

Original post by ghostwalker

There is only one distinct element there. So, one.

Sorry, you'll have to clarify that. How can it be similar and have three elements?

Thinking about it, I'm not entirely sure what I mean either! Thank you for the help regardless.

Reply 7

Zacken

Original post by foorganders

Thinking about it, I'm not entirely sure what I mean either! Thank you for the help regardless.

I suppose you could have a set

\displaystyle \{ \emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}

which has three elements. :smile:

Reply 8

Gregorius

Original post by foorganders

Ah ok, thank you for the clarification. So would the set

\{ \{ \emptyset \}, \{ \emptyset \}, \{ \emptyset \} \}

have three elements? Or one?

As others have already pointed out, the answer is that there is only one element. If you want to consider this object as containing three elements, then you might look at the idea of a multiset.