Math Nerds, Help!

I’m trying to understand the difference between the following operators (not sure if that’s the right term in this case) and when each would be used. Hopefully the math geeks and semantic pedants here can be of help.

What is the difference between:

I know this has to do with sets, but I’m not entirely clear how to visualize the differences between “is a member of” and “Contains” or “Subset of” and Super set of".

This has a lot of them:

I’m not sure what the “backwards” element-of operator is.

Edit: Nevermind, it’s just for listing things backwards. So, x ∈ A is the same as A ∋ x.

The / denotes negation. So x ∈ {x, y, z}, but x ∉ {y, z}.

With regards to membership and containment, it’s a matter of distinguishing between objects and sets. x and {x} are not the same thing, and so we would say that x ∈ {x, y, z} and {x} ⊆ {x, y, z}.

The relation between ⊂ and ⊆ is pretty similar to the relation between < and ≤: the first means “is a strict subset of”, while the second one means “is a strict subset of or is equal to”. So we could say that {x} ⊆ {x, y, z}, and {x, y, z} ⊆ {x, y, z}; but it isn’t true that {x, y, z} ⊂ {x, y, z}.

Note that the “backwards” operators (∋, ∌, ⊃, ⊅) are used far less commonly than the others.

In other words, only a set can be a subset of another set. So you’d only put a set to the left of a ⊂ or ⊄ or ⊆ symbol. On the other hand, a set usually isn’t an element of another set (∈, ∉), though it can be if that set is itself a set of sets.
If the you want a really basic explanation of this set stuff, try this.

For those who like Venn diagrams:

Just a warning: I’ve seen ⊂ used this way, but I’ve also seen ⊂ used in exactly the same way as ⊆. To be better understood, instead of using ⊂, you might want to use ⊆ with a slash through just the underline part (don’t know how to make that symbol for display, though).

Unfortunately, ⊂ is almost always used to mean included in or equal to. It would certainly make sense to use it to mean contained in and not equal to, but mathematicians are not always as logical as the ought to be.

Ok, sorry for the delayed follow up, I got sidetracked over the weekend, but hopefully some of you are still paying attention.

If you have a set of {x, y, z}.

Is {x,y} ∈ {x,y,z} evaluated as true?

Is {x,y} ⊂ {x,y,z} evaluated as true?

Is {x,b} ∈ {x,y,z} evaluated as true?

Is {x,b} ⊂ {x,y,z} evaluated as true?

Is {x,b} ∉ {x,y,z} evaluated as true?

Is {x,b} ⊄ {x,y,z} evaluated as true?

If I read the comment above right, you’d never have the ∈ or ∉ operators with a set on the left side only a single value, or is that mistaken?

It’s on the right conceptual track, but it’s not quite true. The way it works is this:

∈: Set of thingamajigs on the right, a particular thingamajig on the left. It says the item on the left is one of the members of the set on the right.

⊂: On both left and right, there is a set of thingamajigs. It says every member in the set on the left is also a member in the set on the right.

So on the right of ∈, and the left and right of ⊂, you must always have a set.

You don’t have to have a set on the left of ∈, but you can, if what’s on the right of ∈ is a set whose members are themselves sets. (For example, {x, y} ∈ {{a, b, c}, {x, y}, {x, z}}).

No. There are only three true facts ending in " ∈ {x,y,z}". Those three true facts are: x ∈ {x,y,z}, y ∈ {x,y,z}, and z ∈ {x,y,z}.

Yes. The two things in {x, y} are also both in {x, y, z}.

No. There are only three true facts ending in " ∈ {x,y,z}". Those three true facts are: x ∈ {x,y,z}, y ∈ {x,y,z}, and z ∈ {x,y,z}.

No. It’s not the case that both of the things in {x, b} are also in {x, y, z}. Specifically, b is not in {x, y, z}.

Yes. This is just another way of saying “It’s NOT the case that ‘{x, b} ∈ {x,y,z}’ is true”.

Yes. This is just another way of saying “It’s NOT the case that ‘{x, b} ⊂ {x,y,z}’ is true”.

In my case, we will never have a Set of Sets so the statement that a ∈ will always have a single value on the left is apt.

Is there an operator that would evaluate {x,b} as true when compared to {x,y,z}? Intersection or something?

Oh, crap, that was wrong… nevermind.

Well, there are many operators that would (such as, as you say, the “Do these have any element in common?” operator), but most of them have not been considered so ubiquitously useful as to be typically written with a single symbol rather than words (or multiple symbols).

What exactly is your case? Perhaps we can give better advice with more details.

We have some software that uses these operators to compare the values of multi-select pick lists. It was originally implemented incorrectly and I need to define what the new solution should look like and I’m trying to capture all possibilities and make sure I understand the principles right.

One scenario is the user inputs a value, either a single or multiple selection, and that needs to be compared against a different set of inputs (some or all of a different multi-select list). This is pretty straightforward I think. The first value is either going to be a element of or a subset of or not a element of or not a subset of the second set.

The other scenario is that the user picks from a set of items, either one or many, and I need to decide when they’ve selected the right combination of choices. This one is a little tougher for me to get my head around.

Here’s an example:

Suppose we have a selection of choices of pizza toppings in a set.

{Pepperoni, Sausage, Onion, Basil, Peppers, Mushrooms, Olives}

Anytime a combination of these choices contains Pepperoni I need to set Pepperoni to true. Any time Pepperoni, Sausage and Olives is selected I need to set Pepperoni, Olives and Sausage to true. For some reason this is harder for me to grasp and I’m not sure why.

There are people who don’t like Venn diagrams?

I don’t completely understand the example you posted, but I want to comment on this:

In this case, {x, b} is not a subset of {x, y, z}. In more formal notation, you’d write {x, b} ⊄ {x, y, z}.