Question

Are sets just generalised vectors?

Answer 1

I don’t think so. Vectors maintain the ordering property while sets do not. So for example, the vector [-1,2] is a completely different vector than [2,-1]. However the set {-1, 2} and {2, -1} are completely the same set. Furthermore, sets do not allow duplicate values and as such its elements are unique. So set {1, 1, 1, 2} is just {1, 2}. However, vector [1, 1, 1, 2] is not the same as [1, 2] as they have different dimensions. Not sure if any other thing I’m thinking is missing in terms if what distinguishes a vector from a set.

Answer 2

I think you've seen the similarity (vectors and sets are both collections of numbers) and confused yourself into thinking they are intimately linked.



A set needn't be a collection of numbers. It could be a collection of apples or penguins or American presidents.


A set is not ordered. Whereas vectors most certainly are.


The elements of a set are distinct, while the components of a vector can be equal.


A vector is composed of basis-vectors. Different "directions" combined together that can be completely distinct (orthogonal), but can also share some similarity. You might say that components of the vector are specifying quantities for ingredients. In contrast, the elements in sets aren't quantifying amounts of some set number of ingredients, but rather the elements are the ingredients.


The appearance of a vector (the components used to represent it) can be changed by a different perspective. A colour can be thought of as red, green and blue or it can be seen as hue, saturation or value.   Sets just are what they are. If your set is {3,2} there is no perspective from which that set can be {4,1}. The set IS 3 and 2, NOT 3 of something and 2 of something else.


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You can, however generalise sets by adding restrictions and specifications. Generalise enough and you have a "vector space" in which the elements of the set are each vectors in their own right. Vector spaces are the formal way to define vectors, but the vectors are NOT the set or the vector space. The vectors are the elements that live inside the set/vector space.

Answer 3

I'm not quite sure what you have in mind.

Vectors have a lot of algebraic structure associated with them (addition, multiplication with the elements of the scalar field). Sets have none of that. With sets you can only ask whether something is or is not an element of a given set. There is no algebraic structure on sets.

Also, you seem to be thinking of vectors as a tuple of coordinates. That only works well in finite-dimensional case, and if we have agreed on a particular base to represent the vectors in.

Answer 4

I just want to try to collect everything here a bit from all the other comments, give you a few definitions, and hopefully make everything a bit clearer

1) A set is not necessarily a collection of only scalar values. A set is a collection of distinct elements (element means thing). The elements could be real numbers, complex numbers, shapes, letters, names, or anything else you can think of. (There are a few more advanced restrictions, but that is outside the scope here). For instance, I can have the set of all stuffed animals named franklin. Or I could have the set of all words and numbers that start with the letter 'e' in english.

2) A vector is not necessarily a tuple of real numbers. A vector is an element of a set with special properties, called a vector space. A vector space is a set **equiped** with a binary operation (that's to say it has this operation) that is called addition. And it is also defined with scalar multiplication over some field, like the real numbers, which is distributive. One example of a vector space is the one you are thinking of. To use the language I've used here we say that the vector space is the set of points in R^n equiped with addition element-wise and scalar multiplication over the field of real numbers.

3) Something I haven't seen other people mention, but the property of Euclidean space is actually unrelated to the way vectors are defined. Rather, this is a property of these things called manifolds. I, unfortunately, am not so well versed in geometry in order to really dive into manifolds and their Euclidean or non-euclidean characteristics. But try reading around about that if you are interested. A big part if that is how we define the distances between points in our set.

Hopefully that helps sum everything up.

Answer 5

If you change your question to "are sets generalised vector spaces" then the answer is yes, trivially so. A vector space is a set with some additional algebraic structure on it. So if you remove this structure you can recover the underlying set.

Answer 6

Oh boys I hear angry set theorist noises