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

(I) Since R is mapping A to B. Then R must be of the form;

$R=\{(a,b): a\in A, b\in B\}$

But, the first element of R, (v,z) does not obeya this rule. Thus R is not a relation. Hence it's neither injective nor surjective and has no inverse.

(II) Since for every $a, b\in A, ~~ R(a)=R(b)\implies a=b\\ \text{Thus, } R \text{ is injective}\\ \text{Also, every element of B is used up in the relation. Thus, } R \text{ is surjective.}\\ \text{The inverse of R is}\\ R^{-1}=\{(2,1), (3,2), (4,3), (5,4), (1,5)\}.$