Let $n \geq 2$ be an integer and $X=\{1, \ldots, n\}$. Define
$\Sigma=\{\sigma: \sigma \text { is a bijection of } X \text { and }|\sigma(k)-k| \leq 1 \text { for all } k \in X\}$
and for any $\sigma \in \Sigma, Y_\sigma=\{m \in X: \sigma(m)=m\}$. Then

• Correct answer
  A

  number of elements of $\Sigma$ is 1 if $Y_\sigma$ is empty for some $\sigma \in \Sigma$
• Correct answer
  B

  for every $\sigma \in \Sigma, \sigma(\sigma(k))=k$ for all $k \in X$
• Correct answer
  C

  $n$ is odd implies $Y_\sigma$ is non-empty for every $\sigma \in \Sigma$
• Correct answer
  D

  $n$ is even implies number of elements of $Y_\sigma$ is even for every $\sigma \in \Sigma$

Hence, the correct answer is options: 'A', 'B', 'C', 'D'.