Which of the following relations are functions? (i) ${(1,(a,b)),\ (2,(b,c)),\ (3,(c,a)),\ (4,(a,b))}$ (ii) ${(1,(a,b)),\ (2,(b,a)),\ (3,(c,a)),\ (1,(a,c))}$ (iii) ${(1,(a,b)),\ (2,(a,b)),\ (3,(a,b))}$ (iv) ${(1,(a,b)),\ (2,(b,c)),\ (1,(c,a))}$

A relation $R$ is a function iff every first component occurs exactly once. (i) first components $1,2,3,4$ appear once each $\Rightarrow$ function. (ii) $1$ appears twice $\Rightarrow$ not a function. (iii) $1,2,3$ appear once each $\Rightarrow$ function. (iv) $1$ appears twice $\Rightarrow$ not a function.