
Bijective function proof
Hey guys, just a bit lost with this proof
Let x and y be distinct numbers in the set {1, 2, . . . , n}, (n positive integer) and let $\displaystyle \alpha \in$ Sn be the transposition (x y). Define
W = { $\displaystyle \beta \in$ Sn  $\displaystyle \beta$(x) > $\displaystyle \beta$(y) },
Z = { $\displaystyle \beta \in$ Sn  $\displaystyle \beta$(x) < $\displaystyle \beta$(y) }.
Prove that W and Z are complementary subsets of Sn and that f($\displaystyle \beta$) = $\displaystyle \beta\alpha$ defines a bijective function from W to Z.

Hi Mr.
let a = beta(x), b=beta(y).
Then a neq b and 1\le a ,b \le n.
If a < b then beta is in Z. If a < b then beta is in W. Thus
W and Z partition S_n.
Note that beta alpha(x) = beta(y) = b
and beta alpha(y) = beta(x) = a.
Thus,
beta in W
iff
beta(x) > beta(y)
iff
a > b
iff
beta alpha(y) = a > b = beta alpha(x)
\iff
beta alpha in Z
Thus, the map f that takes beta to beta alpha sends
everything in W to Z.
Since f applied twice is the identity, it follows that f also sends
everything in Z back to W.
I.e., f is a bijection between W and Z.
Best,
ZD
