The following exercise of from Guide to Abstract Algebra by Carol Whitehead, 1st Edition 1988.
Let $\bullet $ denote a binary operation on a non-empty set $S$. Suppose that $\bullet $ admits a left identity $e$ and a right identity $f$. Prove that $e = f$.
Although I am a beginner on this topic, I am pretty sure the question is wrong.
One way the question could be correct is if to said the • binary operation was commutative. This would mean:
$$ \begin{align} e \bullet x&=x\\ x \bullet f&=x \implies f \bullet x=x \\ \text{} \\ \therefore e&=f \end{align} $$
Question: Is the original question, as reproduced here, valid?
No, the textbook's assertion is correct. We have $$ e = e \cdot f = f, $$ the first equality since $f$ is a right-identity, the second as $e$ is a left-identity.