Question about Relative Homology I am reading through Hatcher and I came across the following statement and am having trouble making sense of it.  I am not sure why elements may be written this way.  Any help will be appreciated.
By
considering the deﬁnition of the relative boundary map we see:
Elements of $H_n(X, A)$ are represented by relative cycles: $n$ chains $α ∈ C_n(X)$
such that $∂α ∈ C_{n−1}
(A).$
A relative cycle $α$ is trivial in $H_n(X, A)$ iﬀ it is a relative boundary: $α = ∂β + γ$
for some $β ∈ C_{n+1}
(X)$ and $γ ∈ C_n(A).$
 A: The relative chains are $C_n(X,A)=C_n(X)/C_n(A)$. Let $\bar\partial$ be the boundary operator on these reduced chains.
Then $\bar\partial[\alpha]=0\in C_{n-1}(X)/C_{n-1}(A)$ if and only if $\partial \alpha \in C_{n-1}(A)$. Here I am using $[\alpha]$ to denote the coset represented by $\alpha$. 
Similarly, $\bar\partial[\beta]=[\alpha]$ implies that $\partial \beta \in [\alpha]$, which implies $\partial \beta =\alpha+\gamma$ for some $\gamma\in C_n(A)$. (Bear in mind that $[\alpha]=\alpha+ C_n(A)$ as a coset.) It is not hard to see that this implication is actually an "if and only if."
A: The relative chain groups are defined by
$$C_n(X, A) = C_n(X)/C_n(A).$$
The usual boundary map on $C_\ast(X)$ descends to this quotient so that we have a chain complex $(C_\ast(X, A), \partial)$. The relative homology groups of the pair $(X, A)$ are the homology groups of this chain complex.
Now if $\alpha = \partial \beta + \gamma$ for $\beta \in C_{n+1}(X)$ and $\gamma \in C_n(A)$, we have that $[\alpha] = [\partial \beta]$ in the quotient group $C_n(X,A) = C_n(X)/C_n(A)$. Since the boundary map descends to the quotient, we have that
$$\tilde{\partial}[\beta] = [\partial \beta],$$
where $\tilde{\partial}: C_{n+1}(X,A) \longrightarrow C_n(X,A)$ is the boundary map on $C_\ast(X,A)$. Therefore $[\alpha]$ is a boundary in $C_n(X,A)$ with respect to the boundary map $\tilde{\partial}$.
