What is the difference between an element and a subset in set theory?

[daiviko]

I'm attempting to teach myself topology from a textbook. I'm on the first chapter and came into some trouble with some of the set theory.

Here is what the textbook says.

We make a distinction between the object a, which is an elemant of a set A, and the one-element set {a}, which is a subset of A. To illustrate if A is the set {a,b,c}, then the following statements are all correct.

•a is an element of A
•{a} is a subset of A
•{a} is an element of P(A) where P(A) i the power set of A meaning that P(A) is the set of all subsets of A.

However according to the textbook the following statements are not true
•{a} is a member of A
•a is a subset of A

If the set {a}, simply contains a what is the difference between saying a is an "element" of A and a is a "subset" of A? If an object is an element of some set isn't it also a subset of that set? I also am having trouble understanding the idea of a power set. If P(A) is the set of all subsets then doesn't P(A)=A?

 

[jedishrfu]

the textbook is right

{a} is a subset of A and {a} is a member of the PowerSet(A) since the power set contains all subsets of A including ∅ and A itself.

{a} is NOT a member of A.

a is an element of A and a is NOT a member of P(A) as P(A) contains only subsets of A and not any of its elements.

remember a ≠ {a} this is a crucial distinction.

 

[daiviko]

I still don't understand how they are different though. If the only element of {a} is a, then how does a≠{a}?

You also didn't answer my other question. Or maybe you did but I didn't understand it.

The textbook seems to imply that just because a is an element of A doesn't mean a is a subset of A. Could you explain this?

 

[daiviko]

Also, aren't all the elements of A also subsets of A? For example of A={1,2,3}, {1} is a subset of A, right? And wouldn't {1} be one of the elements of P(A)? ugh I'm confusing myself.

 

[jedishrfu]

I still don't understand how they are different though. If the only element of {a} is a, then how does a≠{a}?

You also didn't answer my other question. Or maybe you did but I didn't understand it.

The textbook seems to imply that just because a is an element of A doesn't mean a is a subset of A. Could you explain this?

You can construct sets where the elements are also subset of the set.

consider a set N = { x, {x}, {x, {x} } ... } this is how they sometimes represent natural numbers where x is ∅ the empty set.

sets with in sets within sets. x=0 and {x}=1 and {x,{x}} = 2 ... (see wikipedia: set-theoretic numbers)

but in general the power set contains all possible subsets of A and while members of A could be subsets of A that isn't always true. I mean we could make a set A where some or all of the elements happen to also be subsets of A that's not true in general.

 

[jedishrfu]

Also, aren't all the elements of A also subsets of A? For example of A={1,2,3}, {1} is a subset of A, right? And wouldn't {1} be one of the elements of P(A)? ugh I'm confusing myself.

okay so start with a={1,2,3}: yes {1} is a subset of A and it is a member of the P(A) because by definition the P(A) contains all subsets of A including A itself and ∅ the empty set.

But what you said earlier is that 1 is an element of A but 1 is not an element of P(A) because 1 is not a set.

what the book is saying when it says they aren't TRUE is that they aren't ALWAYS true and MATH really likes to have statements that are ALWAYS true.

 

[dvdl]

[Please don't think I am trying to patronise you here. I still use this kind of explanation to explain university level maths to myself.]

Replace the word 'set' with the word 'bag'.

Now suppose you have 2 bags, one has an apple inside, one has a banana. So you have {a} and {b}.

Now try two experiments:

1) Tip the apple from one bag into the other bag. Now you have a bag with an apple and a banana in it, and an empty bag.

So you have {a, b} and {}.

2) Put one of your bags (which contains an apple) inside the other bag. Now you have one bag which contains a banana and which also contains a bag. And in that second bag there is an apple

So you now have {{a}, b}.

It should be clear from the bag analogy that {a, b} ≠ {{a}, b}.If you understand this, try it with 2 bags and one apple.
Basically, what I'm trying to say is that a set, in the most naive sense, is an object which contains other mathematical objects. Thus a set can contain another set because that second set is a mathematical object. (You can't just pretend the bag isn't there)

An element of a set A is an object which is contained in A.

A subset of a set A is a collection (a set (bag) in it's own right) of elements of A.

 

[dvdl]

In order to understand power sets you need to understand what a subset is.

I have an exercise for you that may help:

Consider the set {a, b, c, d}.

List all the subsets of {a, b , c, d} you can think of.




[A little note. A book on topology will assume prior knowledge of set theory so the introductory chapter on sets will be quite brief. My suggestion to you is that you first find a book devoted to set theory for a more detailed introduction.]