Let $\sigma$ :{1,2,3,4,5} $\to$ {1,2,3,4,5} be a permutation such that: $$\sigma^{-1}(j) \leq \sigma(j) ; \forall j=1,2,..,5 $$ Then ,which of the following are correct?
a) $\sigma o \sigma (j)= j; \forall j$
b) $\sigma^{-1}(j)=\sigma(j);\forall j)$
c) The set {k : $\sigma$ (k)=k} has an odd number of elements.
d) The set {k : $\sigma$ (k) $\neq$ k} has an even number of elements.
I understood here that under the given condition identify permutation definitely holds. So, options a) and b) must be true. But I am unable to list down any other permutation which satisfies the condition given to check the validity of the last 2 options. Please help out.
As already the example of identity shows, c) is false in general.
According to a), $\sigma$ is an involution, hence partitions $\{\,k:\sigma(k)\ne k\,\}$ into pairs $\{x,\sigma(x)\}$. We conclude that $\{\,k:\sigma(k)\ne k\,\}$ has even cardinality, d) is correct. This also shows that c) is in fact always false.