Solve: Composition of relations: A ∘ B
A = {(1, 1), (1, 2), (4, 1), (4, 2)}
B = {(1, 1), (2, 1), (1, 4), (2, 4)}
When I solve mentally I get A ∘ B = {(1, 1), (2,2)}
But when I draw an arrow-diagram I get A ∘ B = {(1, 1), (1, 2), (2, 1), (2,2)}
(1, 2) and (2, 1) from the the arrow-diagram since 4 points to 1 & 2, and 1 also points 1 & 2.
Which one is right?
The diagram is correct.
$B$ maps both $1$ and $2$ to $1$ or $4$. $A$ maps both $1$ and $4$ to $1$ or $2$.
Thus $A\circ B$ maps both $1$ and $2$ to $1$ or $2$.
When you solved mentally, I guess you mistook the relations for inverses which would have meant the composition was an identity. However, as neither is one-to-one, then they are not invertible.