Fixed point/M to M/Intersection of graph and diagonal/Exercise

Let ${}M$ denote a set, and let

$F\colon M\longrightarrow M$

be a mapping. Show that ${}F$ has an fixed point if and only if the intersection of the graph of ${}F$ with the diagonal ${}\triangle ={\left\{(x,x)\in M\times M\mid x\in M\right\}}$ is not empty.

Create a solution