I have some basic questions on relative singular homology. Let $A \subset X$ and let $C_k(X,A)$ be the relative singular k-chains. Then it is said that since $\partial$ carries both $C_k(X)$ and $C_k(A)$ to $C_{k-1}(X)$ and $C_{k-1}(A)$ then $\partial$ descends to a map on the relative chain group. I tried to understand how to explicitly write this map. I like to represent elements of this quotient group as $\sigma^k_{X} + \{\beta^k_{A}\}$ where $\sigma^k_{X}$ is in $C_k(X)$ and so on. Then does the induced map $\partial$ carry $\sigma^k_{X} + \{\beta^k_{A}\}$ to $\partial\sigma^k_{X} + \{\partial\beta^k_{A}\}$ or to $\partial\sigma^k_{X} + \{\beta^{k-1}_{A}\}$. Also is there any conceptual problem with thinking the elements of this relative chain as in the coset form I have written there or is it better to handle them as completely new objects. I will ask my remaining questions depending on the answer to this one so I will keep them for now.
Thanks alot