This question is in reference to the CLRS Introduction to Algorithms book, page 972, Chinese remainder theorem:
On wikipedia, $\leftrightarrow$ is mentioned as the logical equivalence operator. I don't understand what this means in the context of chinese remainder theorem.
As I understand it, an item $[a]_n \in \mathbb{Z}_n$ maps to a unique entry $([a]_{n_0}, [a]_{n_1}, ..., [a]_{n_k}) \in \mathbb{Z}_{n_0} \times \mathbb{Z}_{n_1}\times...\times\mathbb{Z}_{n_i}$.
Is "$a \leftrightarrow (a_1, a_2,...,a_k)$" a way of representing this very idea?
Yes, that is exactly what this symbol means here, bidirectional assignment. In the flow of the statement of the theorem, one has to interpret the first use as declaring the map from $a$ to the remainder tuple, later comes the claim of the theorem that this map is bijective, justifying the double arrow.
The only irritating part may be that the bidirectional assignment symbol is used before the claim of the bijectivity.