Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of functions, weaken it to the classes of partial functions, total relations and general relations and also strengthen it to the class of permutations. Then we investigate the class of involutions and the subclass of proper involutions. Finally, we treat idempotent functions, partial idempotent functions and related concepts. We count relations, calculate corresponding probabilities and also calculate the limiting values of the latter in case that the cardinality of $X$ tends to infinity. All these results have been motivated and also supported by numerous experiments performed with the RelView tool.

翻译：设$X$为有限非空集合，考虑随机选取的$X$上的特定函数与关系。我们分别研究这些函数与关系中不动点与自反点的存在性与不存在性。首先考察函数类，将其弱化为偏函数类、全关系类与一般关系类，同时强化为置换类。接着研究对合类及其子类——真对合类。最后讨论幂等函数、偏幂等函数及相关概念。我们对这些关系进行计数，计算相应的概率，并进一步研究当$X$的基数趋于无穷时这些概率的极限值。以上所有结果均受到基于RelView工具所开展的大量实验的启发与验证。