 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/12/v/solution

If all

n

indices are distinct and each lies in

{1, \dots, n}

, their list is a permutation of

(1, \dots, n)

. Sorting it takes some number

N

(- 1)^{N}

because its sorted component is

1

. Therefore

ϵ_{i_{1} \dots i_{n}} = ⎩ ⎨ ⎧ 0, + 1, - 1, some indices repeat, (i_{1}, \dots, i_{n}) is an even permutation, (i_{1}, \dots, i_{n}) is an odd permutation .

The parity of a permutation is independent of the chosen sequence of transpositions.

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