The following Lemma will be essential in the definition of the determinant.
Lemma A.9.1 There exists a function, sgn_{n} which maps each ordered list of numbers from
 (1.15) 
 (1.16) 
In words, the second property states that if two of the numbers are switched, the value of the function is multiplied by −1. Also, in the case where n > 1 and

 (1.17) 
where n = i_{θ} in the ordered list,
Proof: Define sign

This delivers either −1,1, or 0 by definition. What about the other claims? Suppose you switch i_{p} with i_{q} where p < q so two numbers in the ordered list
Then


The last product consists of the product of terms which were in ∏ _{r<s}

Now consider the last claim. In computing sgn_{n}

and the other terms in the product for computing sgn_{n}

It is obvious that if there are repeats in the list the function gives 0. ■
Lemma A.9.2 Every ordered list of distinct numbers from
Proof: This is obvious if n = 1 or 2. Suppose then that it is true for sets of n − 1 elements. Take two ordered lists of numbers, P_{1},P_{2}. Make one switch in both to place n at the end. Call the result P_{1}^{n} and P_{2}^{n}. Then using induction, there are finitely many switches in P_{1}^{n} so that it will coincide with P_{2}^{n}. Now switch the n in what results to where it was in P_{2}.
To see sgn_{n} is unique, if there exist two functions, f and g both satisfying 1.15 and 1.16, you could start with f
Definition A.9.3 When you have an ordered list of distinct numbers from

this ordered list is called a permutation. The symbol for all such permutations is S_{n}. The number sgn_{n}
A permutation can also be considered as a function from the set

as follows. Let f
In what follows sgn will often be used rather than sgn_{n} because the context supplies the appropriate n.