Thread: Set theory basics

Sorry if this is too basic for uni, but can't find another suitable thread.

Anyway, I'm wondering about the definition of a set as an object that contains other objects, in which all elements must be unique (and their order doesn't matter).

{6, 11} = {11, 6} = {11, 11, 6, 11}

... as Wikipedia puts it.

Do we say that the set {11, 11, 6, 11} has only 2 elements (one of which is repeated twice)?

What would we call the four symbols between the brackets if they're not elements (or members)?

Tokens? Outcomes?

Am I missing a philosophical/logical point here?

Do we say that the set {11, 11, 6, 11} has only 2 elements

Yes.

What would we call the four symbols between the brackets if they're not elements (or members)?

Here, there is a difference between notation and meaning, or syntax and semantics. The meaning of the string "{11, 11, 6, 11}" is an abstract object, set, with two elements. The particular syntactic conventions, such as using curly braces and separating elements with commas, do not exist at the level of meaning. Meanwhile, the concept "element" pertains to the meaning, the abstract mathematical object. So, the two-symbol string "11" that is repeated three times is an incident of notation only and is not directly related to the set as a mathematical object.

Thanks, that clears things up a bit.

So if a probability space S = numbers on a dice
and A is a set of actual outcomes from 10 throws:

S = {1, 2, 3, 4, 5, 6}
A = {3, 1, 5, 2, 4, 3, 4, 5, 3, 6}

then S = A

Otherwise if we threw B:

B = {1, 2, 3, 4, 1, 2, 3, 4, 1, 2}

then B is a subset of S?

For consistency, is A also considered a subset even though it contains all elements of S?
So any 10 throws of a dice will always produce some subset of S?
Have I just answered my own question?

Thanks again

So if a probability space S = numbers on a dice
and A is a set of actual outcomes from 10 throws:

S = {1, 2, 3, 4, 5, 6}
A = {3, 1, 5, 2, 4, 3, 4, 5, 3, 6}

then S = A

Yes, provided A is defined as a set of outcomes. However, this is not the only way to specify the outcomes of 10 throws. For the pupose of calculating probabilities, it may be important to know how many times a particular outcome occurred. One can say that A is a sequence of outcomes, or that A is a multiset of outcomes. Sets, multisets and sequences (tuples) are different mathematical objects. Sequences also usually use a different notation.

Otherwise if we threw B:

B = {1, 2, 3, 4, 1, 2, 3, 4, 1, 2}

then B is a subset of S?

Yes.

For consistency, is A also considered a subset even though it contains all elements of S?

People distinguish proper subsets, which by definition are not equal to the whole set. Otherwise, a set is considered its own (improper) subset.

So any 10 throws of a dice will always produce some subset of S?

Yes.