3

How to express a permutation (without repetition) of a Set $A$?

I'd like to create a set $P$ of tuples while equal tuples should only occur once in the set $P$. Tuples are equal when e.g. $\{a, b\} = \{b, a\}$. Tuples of the same values should not be included, e.g. $\{a, a\}$.

Example: Given $A = \{a, b, c\}$, I would like to receive $P = \{\{a, b\}, \{a, c\}, \{b, c\}\}$.

How can I write this in an equation?

Thomas
  • 191
  • 3
    Can you define exactly what you mean by permutation, the word has acquired several different meanings (though it has one that is more prevalent, and appears to not be what are interested in). – Justin Benfield Jun 26 '16 at 10:27
  • I'd like to create a set $P$ of tuples while equal tuples should only occur once in the set $P$. Tuples are equal when e.g. ${x, y} = {y, x}$. For example: Given $A = {a, b, c}$, I would like to receive $P = {{a, b}, {a, c}, {b, c}}$. – Thomas Jun 26 '16 at 10:59
  • 1
    I am still unclear what the question means despite reading the accepted answer. The answer appears to give you the set of all subsets of $A$ with 1 or 2 elements. But you give two examples in your question which do not fit that. The first has only two of the $n$ possible subsets with one element and with only $n-1$ of the possible $\frac{1}{2}n(n-1)$ possible subsets with 2 elements. Your second example suggests $P$ is unique given $A$ but only gives subsets with 2 elements. Please explain further what you want. – almagest Jun 26 '16 at 20:21
  • Hi @almagest, I've edited my question. Hopefully it's more clear now. I've never really used math that way until a few weeks ago and English is not my first language. Sorry for any confusion. – Thomas Jun 26 '16 at 21:15
  • 1
    Hi @ThomasZuberbühler I thought there might be a language issue and I was worried by the two votes to close (on the basis that the question is unclear). Some people on this site are trigger-happy when it comes to voting to close :) "Permutations" is definitely the wrong word in this context. I think what you want is the set of ALL subsets of $A$ with two elements. Is that correct? – almagest Jun 27 '16 at 04:24
  • Yes that is correct. :) Is there a term for such a subset? – Thomas Jun 27 '16 at 10:46

3 Answers3

2

A concise description of the sets you seek, using set builder notation, are:

$P=\{\{a,b\}:a,b\in A\}$

$P^{\prime}=\{\{a,b\}:a\neq b\textrm{ and }a,b\in A\}$

Edit: In light of comments since posting this, the set you desire appears to be $P^{\prime}$ above, which can also be described as in user3313320's answer.

Justin Benfield
  • 3,992
2

Let $A$ be an $n$-element set. The set of all $2$-element subsets of $A$ is the set $A^{\{2\}} := \{ B \subseteq A: |B| = 2\} = \{ \{a,b\}: a, b \in A, a \ne b \}$. The cardinality of this set is ${n \choose 2}$. But this set does not contain $2$-subsets of the form $\{a,a\}$ which contain repeated elements.

If you want to repeat elements, the objects you are interested in are called multisets. The multisets $\{a_1,a_1,a_2\}$ and $\{a_1,a_2\}$ are distinct (and have cardinalities $3$ and $2$, respectively), although these two multisets are the same set. What you are looking for then is the set of all multisets of $A$ of cardinality $2$.

Ashwin Ganesan
  • 4,045
  • 11
  • 10
0

$P=\{C|C\subset A \text{ and }|C|=2\}$ should work.

In words, it is the set of all subset of $A$ with two elements.

Brian Cheung
  • 1,977