About a set being an element of another set

[d.mehdoi]

Firstly I apologize if it's not the proper category for this question. I have searched sets and found some questions on Advanced Mathematics category so I wanted to ask it here as well. Also there might be some grammar mistakes in my message, I'm not a native speaker.

I'm still learning very basic concepts of Mathematics so my question might sound weird. Lately I have been reading a book about Sets and I saw that a set can be an element of another set, but there is something I could not get. Let's assume we have two sets like this:

A={3, 4} and B={1, 2, 3, 4, {3, 4}}
n(A)=2 and n(B)=5

We can say that set B, along with its elements 1, 2, 3, 4, has set {3, 4} as the fifth element of it. Also we can conclude that A⊆B as B has 3, 4. And there is the fifth element of set B, which is also a set, {3, 4}.

If we were to name that set {3, 4} as D; would that set be typed as D={3, 4} or D={{3, 4}}? What difference would these two ways of typing have if they were shown in a Venn diagram? Is that difference between these two ways of expressions existent only in the context of set B?

If we can type it as D={3, 4}, then isn't A=D? Then how can I show set A inside of B if I were to draw it in a Venn diagram form, different than element {3, 4}. Doesn't it mean we are actually expressing the same elements multiple times?

As far as I could understand, an element can be an element of thousands of different sets in real life. For example, a student in a class can be an element of a set that consists of the students whose names starts with, let's say, J; and at the same time that student can be an element of a set that consists of the students whose school number starts with 3. I mean, going back to D={3, 4} and D={{3, 4}}, is it dependent on the context? How can I choose which one of these expressions I should use to express the same set D when I'm using the listing method?

For this reason, can we say a set can't expressed out of context? Because in this example we have set A which consists of 3 and 4, and likewise set D which also consists of those same elements. Thus they are the same.

If I'm supposed to express set D as {{3, 4}}, and A as {3, 4}; even though they both consist of the same elements, then why this isn't the opposite? Why isn't A expressed as {{3, 4}}, and D as {3, 4}? Why should it be dependent on another set, which is set B in this example?

If set D can be expressed as {3, 4} by itself, just as set A; then I can't imagine how to express set B={1, 2, 3, 4, {3, 4}} in a Venn diagram form because in reality, it will have 3 and 4 two times. If we are supposed to type them once, it's seems wrong to me to do so.

One explanation I could come up with is to see a set like a level of layer. For example lets assume the uppermost layer looks like this
B={1, 2, 3, 4, {3, 4}}
and without thinking deeply, we are supposed to see these 5 elements as they are. So {3 ,4} is nothing but an element, we aren't supposed to think any deeper than that. And as it's an element by itself, we aren't expressing 3 and 4 twice, because actually we aren't expressing them when we say {3, 4}. But when we draw set B={1, 2, 3, 4, {3, 4}} on a Venn diagram, and when we place the element {3, 4} out of set A={3, 4}, it seems a little weird to me. Is there an explanation to this?

Continuing the layer example, if we prefer to go one step deeper, then we will have
{3, 4} and the other elements in the set(1, 2, 3, 4) won't be existent anymore. They were left in the uppermost layer. This one remains because it was a set, too. So a set defined as an element at an upper layer can be a set in the lower/deeper one with its own elements.

I don't know if I could draw that with words but that's what I thought. So eventually there is 3, 4 in the uppermost layer, and another 3, 4 in the lower level of it, which is seen as {3, 4} in this example. As set B is defined in the uppermost layer, we are expected to assume that {3, 4}, which is a set in the lower layer but seen as an element from the uppermost layer where set B is defined, is just an element.

So, am I correct with my conclusion? Is it correct to approach them like they are existent and meaningful only on that level of "layer"? Because otherwise I can't understand placing the element, which is actually a set, {3, 4}; out of set A={3, 4} when set B is drawn in a Venn diagram form.

Thanks

 

[d.mehdoi]

As I'm unable to edit my post, I need to type it here. If there can't be a set as B={1, 2, 3, 4, {3, 4}}, then my question isn't needed to be answered because I will understand it. And I'd like to mention my very limited knowledge in Mathematics once again, so this will probably clarify the weirdness and lack of very fundamental knowledge in my previous question, and in my attempt of explanation.

As I have seen sets like F={1, 2, {2, 3}}, it made me think that if the same element is being an element twice, in different forms, then this kind of set also could be possible, but I'm probably wrong.

Also a note for site admins, it would be nice if they allow users to edit their messages all the time, it's only available for a short time and then you are no longer allowed to edit them. So if you notice a mistake in your message, you are completely unable to edit it. I'll be glad if they can change that.

Thanks.

 

[Steven G]

If we were to name that set {3, 4} as D; would that set be typed as D={3, 4} or D={{3, 4}}. If you named {3, 4} as D, then by your own definition of D, D = {3, 4} .
Now on the other hand if D only contain the element {3, 4}, the D = { {3,4} }

Yes A = D

If I'm supposed to express set D as {{3, 4}}, and A as {3, 4}; even though they both consist of the same elements. What you are saying is NOT true. A has exactly 2 elements, namely 3 and 4 while D has exactly one element namely the set that contain 3 and 4 as its elements.

 

[Steven G]

The reason we place the element {3,4} outside of A is very easy to see (you just need those special glasses) and that reason is as simple as the element {3,4} is NOT an element of A. At this level (only thinking about elements) {3,4} is an element. Period, end of discussion. It is an element in D and 3 and 4 are elements of A. Do not let those curly brackets get to you!

 

[Steven G]

Don't try to recognize what the symbols, ie elements, look like. They are just symbols. The elements in a set are separated by commas and if what is between one set of commas does not look like what is between another set of brackets then the elements are different! Let W = { 3, 4, {3,4}, {{3,4}}, {3, , 4} } has 6 different elements. Can you list them? Please list each element on a different line to avoid any trouble.

 

[Dr.Peterson]

If we were to name that set {3, 4} as D; would that set be typed as D={3, 4} or D={{3, 4}}? What difference would these two ways of typing have if they were shown in a Venn diagram? Is that difference between these two ways of expressions existent only in the context of set B?

If we can type it as D={3, 4}, then isn't A=D? Then how can I show set A inside of B if I were to draw it in a Venn diagram form, different than element {3, 4}. Doesn't it mean we are actually expressing the same elements multiple times?

I don't think you can express sets containing sets in a Venn diagram, which is not designed for that! The circles in the diagram are only groupings; circles inside other circles don't represent sets contained as elements of sets, but merely subsets.

As far as I could understand, an element can be an element of thousands of different sets in real life. For example, a student in a class can be an element of a set that consists of the students whose names starts with, let's say, J; and at the same time that student can be an element of a set that consists of the students whose school number starts with 3. I mean, going back to D={3, 4} and D={{3, 4}}, is it dependent on the context? How can I choose which one of these expressions I should use to express the same set D when I'm using the listing method?

For this reason, can we say a set can't expressed out of context? Because in this example we have set A which consists of 3 and 4, and likewise set D which also consists of those same elements. Thus they are the same.

If you've defined D as {3,4}, then you can call it either D or {3,4} anywhere. In fact, you could say that B={1, 2, 3, 4, D}. And since you've also defined A as {3,4}, you can write A = D.

I think of the braces as the paper bag containing the elements (or maybe a cell membrane containing the cells contents); it turns the elements into a single entity, so that putting {3,4} into a set is not the same as putting 3 and 4 separately into the set. Again, the Venn diagram can't show this, unless maybe you add some new feature to it.

If I'm supposed to express set D as {{3, 4}}, and A as {3, 4}; even though they both consist of the same elements, then why this isn't the opposite? Why isn't A expressed as {{3, 4}}, and D as {3, 4}? Why should it be dependent on another set, which is set B in this example?

If set D can be expressed as {3, 4} by itself, just as set A; then I can't imagine how to express set B={1, 2, 3, 4, {3, 4}} in a Venn diagram form because in reality, it will have 3 and 4 two times. If we are supposed to type them once, it's seems wrong to me to do so.

I think you need to explain why you think D has to be expressed as {{3,4}}; did someone tell you that? What do you perceive to be the difference between A and D as you have defined them? Also, do you mean something special by making some elements blue?

I suspect that the Venn diagram idea may be what's messing you up. Venn was working with logic, not set theory, when he invented the idea, and it really is not general enough to cover this. So much of your thinking is based on a misunderstanding.

One explanation I could come up with is to see a set like a level of layer. For example lets assume the uppermost layer looks like this
B={1, 2, 3, 4, {3, 4}}
and without thinking deeply, we are supposed to see these 5 elements as they are. So {3 ,4} is nothing but an element, we aren't supposed to think any deeper than that. And as it's an element by itself, we aren't expressing 3 and 4 twice, because actually we aren't expressing them when we say {3, 4}. But when we draw set B={1, 2, 3, 4, {3, 4}} on a Venn diagram, and when we place the element {3, 4} out of set A={3, 4}, it seems a little weird to me. Is there an explanation to this?

I agree with what you say here; as far as this B is concerned, {3,4} is just an element. Again, don't try to do this with a Venn diagram!

So, am I correct with my conclusion? Is it correct to approach them like they are existent and meaningful only on that level of "layer"? Because otherwise I can't understand placing the element, which is actually a set, {3, 4}; out of set A={3, 4} when set B is drawn in a Venn diagram form.

I don't think your idea of layers is helpful, as it seems too rigid. Sets contained in sets can be thought of as sort of a hierarchy, but not so that everything at a certain level somehow belongs to that level.

But I do think that forgetting about Venn diagrams in this context will be helpful.

Also a note for site admins, it would be nice if they allow users to edit their messages all the time, it's only available for a short time and then you are no longer allowed to edit them. So if you notice a mistake in your message, you are completely unable to edit it. I'll be glad if they can change that.

When people are allowed to edit old posts, it can really mess up conversations, as things that have been commented on get changed and no one can tell what was really said unless it was quoted. It's like allowing time travel in physics! If the system could be changed so that edits can be done until someone has responded, that might work, but it's probably not an option. I wouldn't mind a little longer than half an hour as an editing window, but on the whole, I think it's best when conversations work like in real life, where you can't unsay something, but just have to say "Excuse, me, but this is what I meant to say ..."

 

[d.mehdoi]

First of all I'm very thankful for your replies, Jomo and Dr. Peterson. Thanks for these very detailed explanations to my question.

I think I see where I have made the mistake now. In the part of the book where I have read that a set can be an element of another set, the example given was something like this:
A={{4, 5}, 3, {2, 1}} and set {4, 5} is an element of set A. s(A)=3

This made me consider {4, 5} as a set with two elements, being an element of A and a set on its own with two elements at the same time. I was confused, and my conclusion was wrong.


In my previous post, I have used blue color to distinguish the other elements from the element {3, 4} because there were elements like 3, 4, {3, 4}. There was no other particular reason for using that color.

And thanks for the explanation about the reason why users aren't allowed to edit/delete posts after 30 minutes.

 

[Dr.Peterson]

I think I see where I have made the mistake now. In the part of the book where I have read that a set can be an element of another set, the example given was something like this:
A={{4, 5}, 3, {2, 1}} and set {4, 5} is an element of set A. s(A)=3

This made me consider {4, 5} as a set with two elements, being an element of A and a set on its own with two elements at the same time. I was confused, and my conclusion was wrong.

But {4, 5} is a set with two elements. How could it not be? It is also an element of A.

This is no different from 3 being both a number and an element of A. There is no conflict.

I assume your notation s(A) means the cardinality (size?) of set A. I more often see that as n(A), as you've used before, or |A|. Notations vary!

 

[d.mehdoi]

But {4, 5} is a set with two elements. How could it not be? It is also an element of A.

This is no different from 3 being both a number and an element of A. There is no conflict.

I assume your notation s(A) means the cardinality (size?) of set A. I more often see that as n(A), as you've used before, or |A|. Notations vary!

Hello. Thanks for your reply Dr. Peterson.

Let's assume we have added two new elements to the set, 4 and 5.

Now we are going to have A={{4, 5}, 3, {2, 1}, 4, 5}. I have made the newly added ones blue, even though they are already distinguishable. Because I might confuse them.

If {4, 5} is already a set with two members, as you have mentioned, let's name it D. And we'll have D={4, 5}. I hope I didn't make any mistakes so far.

Now let's assume our other 4 and 5 are also members of another set. Set G, let's say. G={4, 5}.

As a result, we have D={4, 5} and G={4, 5}; doesn't it make them the same? Thanks to your and Jomo's replies, I understand that {4, 5} and 4, 5 aren't the same, but if we express them as D={4, 5} and G={4, 5}; we have D=G as a result. And if D=G; then when we type these sets as letters in set A it'll be like:
A={D, 3, {2,1}, D} or A={G, 3, {2, 1}, G};
and as we aren't supposed to type the same element twice, it'll be either A={D, 3, {2, 1}}; or A={3, {2, 1}, G}.

This is why I thought there could be another way of expression, because we are getting the same thing if we express them both as {4, 5}, which consequently make the first set A seem like having the same element twice, even though they aren't the same, considering the difference in their ways of expressions({4, 5} and 4, 5).

I don' t understand how I can differ D={4, 5} and G={4, 5} when they are defined separately.

 

[Dr.Peterson]

As a result, we have D={4, 5} and G={4, 5}; doesn't it make them the same? Thanks to your and Jomo's replies, I understand that {4, 5} and 4, 5 aren't the same, but if we express them as D={4, 5} and G={4, 5}; we have D=G as a result. And if D=G; then when we type these sets as letters in set A it'll be like:
A={D, 3, {2,1}, D} or A={G, 3, {2, 1}, G};
and as we aren't supposed to type the same element twice, it'll be either A={D, 3, {2, 1}}; or A={3, {2, 1}, G}.

Yes, your set G is not "another set"; it is the same set as D.

But, no, the set A={{4, 5}, 3, {2, 1}, 4, 5} is not the same as A'={D, 3, {2,1}, D}, because 4 and 5 are not elements of the latter.

Observe that D is a subset of A (because 4 and 5 are both elements), and also an element in itself.

Please be aware that this sort of thing is not normal; we don't usually talk about sets as odd as that! But we have to define concepts clearly so that we can talk about such things when needed. One place where such things do show up is in defining numbers in terms of the empty set: 0 = {}, 1 = {{}}, 2 = {{},{{}}}, etc.

This is why I thought there could be another way of expression, because we are getting the same thing if we express them both as {4, 5}, which consequently make the first set A seem like having the same element twice, even though they aren't the same, considering the difference in their ways of expressions({4, 5} and 4, 5).

I don' t understand how I can differ D={4, 5} and G={4, 5} when they are defined separately.

The answer is that they are not distinct, so you can't distinguish them. (That's also why I don't accept your coloring some 4's blue, as if they were not the same.)

 

[d.mehdoi]

Hello and thanks for your reply, Dr. Peterson.

I see that that "secondary step of thinking" isn't necessary in defining a set and its elements. I don't know if it's a proper way of naming it but by "secondary step" I mean A={3, 4, {3, 4}} for example here, that {3, 4} being the same doesn't mean much while defining set A. We should consider set A as is.

One place where such things do show up is in defining numbers in terms of the empty set: 0 = {}, 1 = {{}}, 2 = {{},{{}}}, etc.

Thanks for the link, it's very interesting.

I assume your notation s(A) means the cardinality (size?) of set A. I more often see that as n(A), as you've used before, or |A|. Notations vary!

I apologize for forgetting to reply this question in my previous post. It should have been n(A), my bad. s in s(A) stands for, I'm not sure but I guess, "Sayı", which means number in Turkey. I'm used to type s(A), too. This is why I have confused them in my message.

 

[Dr.Peterson]

I see that that "secondary step of thinking" isn't necessary in defining a set and its elements. I don't know if it's a proper way of naming it but by "secondary step" I mean A={3, 4, {3, 4}} for example here, that {3, 4} being the same doesn't mean much while defining set A. We should consider set A as is.

That's a good way to think about it: To set A, {3,4} is just another element; it "doesn't open the bag and look inside"!

 

[d.mehdoi]

That's a good way to think about it: To set A, {3,4} is just another element; it "doesn't open the bag and look inside"!

Yes! I should see it that way.

Thanks.