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)

Part 1

$A=\{v,w,x,y,z\} ; B=\{1,2,3,4,5\}\\ R= (v,z), (w,1), (x,3) (y,5)$

R is not a function as $a \in A$ but $z \not \in B$

So, (v,z) can not be possible for a function

Neither injective nor subjective function and no inverse exists

Part 2

$A = \{1,2,3,4,5\}, B=\{1,2,3,4,5\}\\ R = \{(1,2),(2,3),(3,4),(4,5),(5,1)\}$

Then as for every $a \in A$, there is unique $b \in B$ such that $(a,b) \in R$

$\implies$ R is subjective

The inverse of R exists

$R^{-1}= \{(2,1),(3,2),(4,3),(5,4),(1,5)\}$