Set Theory Questions - Member of vs. Subset of

Factual Questions
1

Was hoping the mathematicians one the dope could explain the difference between the following set operators to me.

∈ versus ⊂; aka “Member of” vs. “Subset of”

What’s the difference between these two concepts? As far as I can tell, if A = {a, b} and B = {a, b, c}, then A ∈ B and A ⊂ B are both true.

What’s the difference?

As a follow on question.

If you want to create a formula where A is {a, b, c, d, g, i} and B = {c} it’s evaluated as true when A is selected (left side) and B is the comparison (right side) is there a better way to do this than just reversing the set operator symbols? Also what happens when A is {a, b, c, d, g, i} and B = {c, h}? Is there a operator for that to be true?

And if you were to do all these things using plain English instead of the special characters what would be language be to be mathematically/programmatic accurate?

2

No. Only A ⊂ B is true.

The only items which are ∈ B are a, b, and c. So a ∈ B, b ∈ B, and c ∈ B are true, but anything else ∈ B is false.

A ⊂ B means “Each item in A is also an item in B”. This only makes sense when both A and B are sets. In your example, it is true because A has two items (a and b), and both of those are also items in B.

A ∈ B means “A itself is one of the items in B”. This also only makes sense when B is a set. However, A needn’t be a set here. Indeed, usually, it won’t be (most often, we are interested in sets whose items are not themselves sets). In your example, this is false because B has three items (a, b, and c), and A is not one of those three items.

3

WARNING: Arcane minutiae ahead, not a topic of concern for introductory students

On the other hand, though, if you get right down to the foundations of math as currently practiced, everything is a set. And in the standard set-theoretical construction of the natural numbers, every element of the set of naturals is also a subset of the set of naturals. Specifically, 0 is the empty set, 1 is the set containing 0, 2 is the set containing 0 and 1, and so on, with each number being represented by the set of all previous natural numbers.

4

So why did you post it?

5

I have no idea what you’re saying here. There are many operators which will take any particular pair of sets to true. I get the impression you have some particular intention in mind, but I don’t know what that intention is.

You tell me. Use plain English to say what you’d like to say, and if there’s anything misguided in it, I’ll let you know. But generally, I trust your competence with your native language.

If this question is about the plain English translation of the two particular symbols you used, just remember, “a ∈ B” stands for “a is an item in B”, and “A ⊂ B” stands for “Each item in A is also an item in B”.

6

Should we only discuss things of concern to introductory students? A warning seems the appropriate way to sidebar other discussion.

7

(I will say that “Everything’s a set!” is about as useful a way of looking at mathematics as “Everything’s a topological space!” or “Everything’s a function!” or what have you. You can make up encodings and identifications to cram any particular piece of mathematics into that ontology, but why would you want to? (It’s natural enough for ordinal numbers, but it’s already grotesque once one discusses ordered pairs. Not to mention suggesting we can’t discuss, say, the set of posters in this thread till we’ve identified its four elements as sets of sets of sets…). Still, sigh, mathematicians do occasionally say such things, even if it has nothing to do with how they actually work and think.)

8

“Ouch!” said Shea. “That’s almost as bad as Frege’s definition of number.”
Bayard droned: " ‘The number of things in a given class —’ "
“Stop it, Walter! It drives me nuts!”
" ‘— is the class of all classes that are similar to the given class.’ "

L. Sprague De Camp and Fletcher Pratt: The Incompleat Enchanter.

i.e., yes, zero “is” the empty set, etc.

Conventionally, an “item” as a member of a set is at an irreducible level; an orange is a member of the set of fruits. “The Citrus Fruits” would not be a member, because it’s a complex thing, thus reducible. And yet “The Citrus Fruits” could be an item, a member of the set of “Kinds of classes of fruits.”

Still, one would generally not say that “The Chess Club” was a “member” of the “High School Class of '74,” even if all of the members of the Chess Club were.

9

Indistinguishable mentions it in Post #2 above, but maybe we could elaborate on this: It is possible (although not necessarily typical) for a member of a set to be another set. Thus, one could have:

A = { x, y }
B = { a, b, c, { x, y } } or equivalently B = { a, b, c, A }
Then you could certainly say that A ∈ B because the set { x, y } is itself a member of B.

When a set is a member of another set, that member set is just one member. If that member is a composite object (having multiple components itself), those components are members of the member, but not members of the “outer” set. (Got that?) Here, x and y are members of A, but they are not members of B.

Question: What is the size of set B? That is, how many elements does it have?
Answer: 4 elements, namely: a, b, c, and { x, y } where { x, y } counts as just one member of B.

(n.b., I use the words “member” and “element” as being totally synonymous and interchangeable. I tend to prefer “member” myself. The word “element”, to me, has a connotation of “atomic, not being composed of smaller parts”. But the set { x, y } is a composite object, being made up of parts. Thus I call it a “member” of B, but I have a bit of a problem calling it an “element”.)

Chronos wants to inundate us with arcane minutiae. How about this to wrap your brains around: If a set can contain a set as a member, can a set contain itself? ( Indistinguishable, Chronos, and Trinopus, you know damn well where that question is going! STFU and let Omniscient think about it a while! :smiley: )

10

Okay, let’s give this a try. I think the phrases you need to know are “is a subset of” and “is a superset of”. (Pause while I dig through my character map for the right symbol here . . . )

Okay: ⊂ means “is a subset of” while ⊃ means “is a superset of”

If A = { 1, 2, 3, 4, 5, 6 } and B = { 2, 4, 6 } then we have:
B ⊂ A (B is a subset of A), and A ⊃ B (A is a superset of B).

So when you ask: Is there a better way to do this than just reversing the set operator symbols? I think the answer is: No, that is exactly how you do it. Use the ⊃ symbol and read it out loud as:
“The set A = {a, b, c, d, g, i} is a superset of B = {c}”

As for the case A = {a, b, c, d, g, i} and B = { c, h } about the best you can say is:
– Neither is a subset of the other. Their intersection is non-empty (that is, they have some members in common), and their union is a bigger set than either alone (that is, each has at least one member not in the other, so that each set contributes something to the union that the other doesn’t).

ETA: See this thread, and particularly Post #20, for another interpretation of the meaning of “member”.

11

I don’t understand why his is a question, except to show off you know how to get the member and subset symbols.

A = set of whole numbers. (0,1,2,3,4,5,…}
B = set { 1,2,3}
the elements of B are also elements of A, but B is not an element of A; however it is a subset.
Sets can contain anything as elements, they can even include other sets as elements - but a set does not by definition include as elements all its subsets.

A subset CAN be an element, but it does not HAVE TO be one.

This was in the first reply.

12

…especially from someone who’s Omniscient. To me, the distinction between an element and a (sub)set of elements seems clear-cut and intuitively obvious, though I can’t even remember when I first learned about sets. And it can admittedly get confusing when you start talking about sets that are elements of other sets (hence, things like Russell’s Paradox), but there’s nothing like that in the OP’s example.

13

Here’s a question about something you may not yet be clear about. If so, then getting clear on it may help you see why it is false that A ∈ B in the quote above.

Take the set Q = {{a, b}, c}.

Here’s a question. How many members does Q have?

The answer is:

2.

Does this surprise you, or did you already know this?

14

Well, if OP read Post #9 he might be able to figure that out.

15

Also, since you’re asking about notation, note that ⊂ is the notation for a “proper” subset. It’s easier to introduce this in terms of an “improper” subset (⊆).

A ⊆ B if every member of A is also in B (i.e. {a,b} ⊆ {a,b}; {a,b} ⊆ {a,b,c} etc). A proper subset, is similar except A also cannot be the same set as B. {a,b} ⊄ {a,b}.

16

{a, b} is not in {a, b, c}? Is there a distinction between {a, b} and a and b that I’m missing? Does {a, b} ∈ {a. b, c} mean something different than "is a and b a member of {a. b, c}?

You can have a set of one, correct? So for every case you can use {a, b} ⊂ {a, b, c} or {a} ⊂ {a, b, c} to accomplish the same thing as {a} ∈ {a, b, c}, correct?

Basically you’re saying {a} AND {b} is not the same thing as {a, b}. So {{a, b}, c} is different than {a, b, c}. Can you explain why this is because it isn’t registering with me.

17

{a, b} means “the set containing a and b” or “the set whose members are a and b.” {a, b, c} is a set with three members. Those three members are a, b, and c. Notice that none of those three members is itself a set. Thus, “The set containing a and b” is not one of the members of {a, b, c}.

{{a, b}, c} is a set with two members. One of those members is {a, b}; the other one is c.

{a, b, c} is a set with three members. Those members are a, b, and c.

18

{a} is the set containing “a”, in notational terms:

a ∈ {a}

“a” is a member of the set containing “a”. However, “a” is not a set, a ⊄ {a} because a is not a set at all, it’s an object, a member of set.

To explain the distinction between {a,b,c} and {{a,b},c}, I’m going to do something crazy and introduce power sets, because I think it illustrates the point well.

P(A) – the power set of A, is the set containing all (improper) subsets of A.

P({a,b}) = {{},{a},{b},{a,b}}

This can’t be {a,b,a,b}, because sets can’t have redundant elements – it would be {a,b}. If you could get rid of those brackets, P(A) would just be A! There would be no way to have a “set of all subsets” if you could could just take the members out and have it be the same.

Maybe I just confused you further, but I think it’s a good way to illustrate why they’re different at least.

19

Trying to translate user stories in technical specs. It’s a programming problem, basically. The end user inputs a set of values and those values are compared to another set of values defined by the admin. The admin defines the operator used to compare those sets changing when the output is true or false.

The simplest and most common case is 2 values, equals versus not equals. This case I’m struggling with is when the input and the comparison are a set of choices instead of 1 value, and I need to understand the different set operators that can be used and how they’ll evaluate predictably for the many different possibilities.

I’ll wait until you read the above description before responding here, I think that may get the point across. The main issue is that we don’t have the flexibility to change which side of the equation the user input set and the admin defined comparison set are on, so properly explaining the function is trickier.

20

Yes, there is a distinction between {a, b} and a and b, as described in several posts above by me and others.

“a” is a “thing”. “b” is a “thing”. These “things” can be collected together into a collection, also known as a “set”.

But a “set” as a whole is also a “thing” and even that can be included as a member in (another) set.

So the “set” { a, b, { y, z } } contains three “things”: The thing “a”, and the thing “b”, and the thing { y, z }, where the thing { y, z } in turn contains the things (sub-things?) “y” and “z”. But the things “y” and “z” are not (individually) considered to be parts of the thing { a, b, { y, z } }

A set can contain a single thing (member). Still, the set as a whole is distinct from the thing it contains. Thus, {x} is distinct from x. You could have an object x, and a set {x} containing the object x, and a set { {x} } containing the object {x}.

ETA: For the same reason, it is emphasized that the empty set, commonly represented by the symbol ⌀ must never be written as {⌀} because that is a non-empty set containing ⌀ as a member!