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The way I understand the definition of a combination of a set is that, without repetition,for example, the combination of the set $\{1,2\}$ would be $(1,2)$, and not $(1,1)$, $(1,2)$ and $(2,2)$ because I thought I read that each element has to be unique. Is this the case?

Also,

What is the combination of a set that has only one element? like $\{2\}$ ?

the empty set? or is it $(2,2)$ which would to violate the unique property, if uniqueness actually is a constraint. I'm not sure if it is or not because the web doesn't cover the topic well in a 'non mathematician can understand it' way.

Teddy38
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3 Answers3

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Usually when we talk about combinations, we are talking about choosing $k$ elements from a set with $n$ elements. Your example seems to illustrate this for $k=2$.

If you think about it this way, it doesn't really make sense to choose 2 elements from a set with only 1. This is why we require $k<n$.

David Kleiman
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You're right -- uniqueness is necessary. So there are no pairs $(x, y)$ that you can make out of the set $\{2\}$.

(As a side note, what you call the "combination of a set" is more commonly called something like "all combinations of two elements" of a set).

Eli Rose
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A k-combination of a set S is a subset of k distinct elements of S.

Consider your set $S = \{1,2\}$.

The 1-combinations are the sets $\{1\}$ and $\{2\}$. The only 2-combination is the set $\{1,2\}$. There are no k-combinations of the set for $k>2$.

For the one element set $S = \{2\}$, there is only a 1-combination, the set $\{2\}$.

As pointed out in the comment below, all sets also have a 0-combination: the empty set.

Adam Francey
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