How can we determine the number of elements in a set?

[Kyouran]

I have the following problem:

Suppose I have a set called "the white chess pieces on a chess board". I could say the set contains the following: king, queen, rook, bishop, knight, pawn. This means the set contains 6 elements. I could also say the set contains: king, queen, 2 rooks, 2 bishops, 2 knights and 8 pawns. Hence, the set contains 16 elements. This is because I now see the 8 pawns, 2 bishops, 2 rooks and 2 knights as being different elements. You could say I gave them an additional property.

So here is my question: Can we actually speak of "the number of elements in a set"? If I have a set of natural numbers between 2 and 5, I could include the number 4 more then once, by adding an additional property let's say colour. I could have a "red" number 4 and a "green" number 4. Now I realize that numbers are abstract and colour isn't, but I could also think of a more abstract property to differentiate between number 4's (like I did with the chess pieces).

 

[HallsofIvy]

I have the following problem:

Suppose I have a set called "the white chess pieces on a chess board". I could say the set contains the following: king, queen, rook, bishop, knight, pawn. This means the set contains 6 elements. I could also say the set contains: king, queen, 2 rooks, 2 bishops, 2 knights and 8 pawns. Hence, the set contains 16 elements. This is because I now see the 8 pawns, 2 bishops, 2 rooks and 2 knights as being different elements. You could say I gave them an additional property.

So here is my question: Can we actually speak of "the number of elements in a set"? If I have a set of natural numbers between 2 and 5, I could include the number 4 more then once, by adding an additional property let's say colour. I could have a "red" number 4 and a "green" number 4. Now I realize that numbers are abstract and colour isn't, but I could also think of a more abstract property to differentiate between number 4's (like I did with the chess pieces).

Once you have defined the set, yes. If you define a set as containing the white chess pieces, it contains 16 elements. If you, instead, define the set of "distinct kinds of chess pieces" then it contains 6 elements, the King, Queen, Bishop, Knight, Rook, and Pawn. But those are NOT "different numbers of elements in the set", those are different sets.

If you define a set to be "the set of natural numbers between 2 and 5 (not inclusive)" that set contains 2 members, "3" and "4". Sets (as opposed to "mult-sets) do NOT include several copies of the same thing so you can't add more "4"s. Different colors don't change that because your definition of the set said nothing about color- that's not part of the set definition so not part of the set. And, as you say, it doesn't make sense to talk about the 'color" of a number. You might talk about the numeral "4", written in red ink or blue ink, but again, whether those would count as "different" would depend on exactly how you had defined the set.

 

[CRGreathouse]

So here is my question: Can we actually speak of "the number of elements in a set"?

Yes.

You might consider comparing the notion of "set" to those of "multiset" (sometimes called "bag") and "tuple". Sets don't allow duplicates ({1} = {1, 1}) and don't consider order ({1, 2} = {2, 1}); multisets allow duplicates ({1} ≠ {1, 1}) and don't consider order ({1, 2} = {2, 1}); tuples allow duplicates ((1) ≠ (1, 1)) and consider order ((1, 2) ≠ (2, 1)).

 

[CRGreathouse]

Also possibly relevant is the concept of an equivalence class.

 

[Kyouran]

The problem with multisets is that they consider duplicates, while this isn't what I meant. What I meant were distinct elements.

If you have a collection of animals, 2 elephants are different from each other. One may be a bit bigger than the other, one might have his tusks removed (damn ivory hunters:p), etc. So they aren't exactly duplicates.

My maths teacher used to say you couldn't add two things which were different (like adding apples to oranges). But I find that in the real world, nothing is equal. Yet, even though they are different, we are able to do maths with these elephants: 1 elephant and 1 elephant yields 2 elephants, whether the first one was bigger then the other or not.

 

[Tac-Tics]

You're hitting on a pretty fundamental principle of math. It's the answer to the seemingly simple question, "when are two things equal."

Sometimes, equal things are obvious. 1 = 1. No one really argues. 1 + 1 = 2. Yeah, easy. Your mom is your brother's mom. Simple.

But what about the two astrological objects commonly known as the Morning Star and the Evening Star? They happen to both be the same object (http://en.wikipedia.org/wiki/Venus). But it isn't without a little knowledge of astronomy that you would know that. Or that the area under e^(-x^2) is the square root of pi. Those things are non-obvious. Equality isn't obvious. It generally requires a proof.

The equality of mathematical functions was one of the driving motives for early computer science. When are two functions "equal"? Mathematically, two functions f and g are "equal" when f(x) = g(x) for all x (and the domains are the same). So we know that f(x) = 2(x + 1) is equal to g(x) = 2x + 2. But are they the same? Well, not really. They are defined differently. If you execute both on a computer, they will take different times. They have different names, f and g. Clearly, they are not always interchangeable.

How much information is associated with the entity? Is it just a rook? Or is it a white rook? Or is it the white queen's rook? Is it the rook at 5d? (The board position is important). Is it the rook that got captured two turns ago (The game history is important). Is it the rook that cost you the game at the regional championship? (Your personal history is important).

How much of that information is relevant to the problem you're trying to solve? Only then can you figure out how big your set is. You can't count things doubly, and a set only has one cardinality. It either has 6 elements or 8, not both. But how many it has depends on how you modeled the problem.