Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2,3),(3,4),(4,5),(5,1)

Let $A = \{1,2,3,4,5\}, B=\{1,2,3,4,5\}.$

Let us determine whether the relation $R = \{(1,2),(2,3),(3,4),(4,5),(5,1)\}$ is an injective function. Since there are no different pairs with the same second coordinate, we conclude that the relation is an injective function.

Taking into account that $range(R)=\{b\in B\ |\ (a,b)\in R\}=\{2,3,4,5,1\}=B,$ we conclude that the relation $R$ is a surjective function.

It follows that the inverse of $R$ exists and it is equal to $R^{-1}=\{(b,a)\ |\ (a,b)\in R\}=\{(2,1),(3,2),(4,3),(5,4),(1,5)\}.$