How do i write "The set of sequence B contains all possible order of item in the set A"? [closed]

Suppose I have a set $A$:
$$A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$

And then I want to have a set of sequence $B$ which contains all possible order of item in the set $A$ , how do I write it?

Solution 1:

When you say "all possible orders" of items in the set $A$, I assume you are referring to permutations, and I'm assuming you are only looking at permutations of ALL elements of $A$, i.e. permutations of length $9$, i.e. rearrangements or bijections on $A$. In other words:

$$B=\big\{ f: A\rightarrow A \mid f \text{ is a bijection} \big\}$$

In that case, this can be written as $\text{Bij}(A)$. It can also be called the "set of symmetries" of $A$, denoted $\text{Sym}(A)$.

Given the simplicity of the set you have chosen (it's just 9 elements and the names are arbitrary), we can also write this as $\text{Sym}(9)$ or even $S_9$ if we want to emphasize the structure of function composition and the fact that it forms a group (called the symmetric group). Some folks prefer to write $S_{9!}$ in order to emphasize the number of elements in the group.

Let me know if I misunderstood your question, and I will gladly edit my response accordingly.