Question

Let A = {a,b,c,d}, B = {1,2,3}, and R = {(a,2), (b, 1), (c, 2), (d, 1)}.

(a)Is R a function?

(b)Is R−1 a function?

Explain your answers.

Answer 1

(a) A function $f$ from $A$ to $B$ is an assignment of exactly one element of $B$ to each element of $A.$

Yes, $R$ is a function: for each input there is exactly one output. 

(b) $R^{-1}$ is not a function. For example, $b \not= d,$ but $f^{-1}(1)=b, f^{-1}(1)=d$