Discrete Mathematical Structures by Pearson

7-24 Discrete Mathematical Structures

1. All the factors in A, which contain neither x

i

nor x

j

will remain unaltered.

2. The factor (x

i

- x

j

) is changed to its negative (x

j

- x

i

).

3. The factors that contain either x

i

or x

j

can be grouped into pairs of products (x

k

- x

i

)(x

k

- x

j

),

where k ≠ i or j, which remain unaltered when x

i

and x

j

are interchanged.

Hence, A operated upon by a transposition t will give -A.

Now, the permutation P considered as the product of a transpositions when operated on the

alternating function A will give (-1)

a

A. Similarly, the permutation P when considered as the

product of b transposition will give (-1)

b

A.

Therefore, (-1)

a

A = (-1)

b

A

or (-1)

a

= (-1)

b

that is, either both a and b are odd or both are even.

Hence, a