Extensions and Pushouts using an exact sequence of sets

Question

This might seem a strange way of doing things, that is, inventing a possible example (according to comments, there is no such thing as an exact sequence of sets), but let us try to make one for exemplary purposes. The questions are at the bottom.

EDIT:

As noted in a comment, there are no short exact sequence of sets and so I changed this to a long exact sequence of sets as we don't have to worry about the initial and final $\rightarrow$'s, only a finite set of exact intermediate terms.

We let all the morphisms $f$ (i.e. $\alpha,\beta$ etc.) acting on any set $S$, by acting element wise and for $x \in S$ if $x \in im(f)$ then $f(x) = x$ and $0$ otherwise.

\begin{array} \mathrm{\cdots } &\longrightarrow & \{B,C,D,F\} & \overset{i_1} \longrightarrow & \{B,C,D,E,F,H,O,N\} & \overset{\pi_1} \longrightarrow& \{E,H,O,N\} &\longrightarrow&\cdots \\ & & \downarrow & &\downarrow & & \downarrow & &&\\ \mathrm{\cdots} &\longrightarrow & \{A,B,D,M\} & \overset{i_2} \longrightarrow & \{A,B,D,E,H,M,N,O\} & \overset{\pi_2} \longrightarrow & \{E,H,O,N\} &\longrightarrow &\cdots \end{array} where

$\alpha:\{B,C,D,F\} \rightarrow \{A,B,D,M\}$

$\eta: \{B,C,D,E,F,H,O,N\} \rightarrow \{A,B,D,E,H,M,N,O\}$

$\beta: \{E,H,O,N\} \rightarrow \{E,H,O,N\}$

Now to show the concept of a pushout, that if another set say $\{A,B,D,K,O,H\}$ in which:

$\alpha ':\{A,B,D,M\} \rightarrow \{A,B,D,K,O,H\}$ and

$\eta ':\{B,C,D,E,F,H,O,N\} \rightarrow \{A,B,D,K,O,H\}$

$\rho :\{A,B,D,E,H,M,N,O\} \rightarrow \{A,B,D,K,O,H\}$

We find that $\eta \circ i_1 = \{B,D\} = \alpha \circ i_2$ But also checking $\eta ' \circ i_1 =\{B,D\} = \alpha ' \circ \alpha$

Looking at the diagram from here commutative diagram for a pushout Here is the diagram I am interested in:

Then $\{A,B,D,E,H,M,N,O\}$ should be the pushout!

The questions are:

1.I observe: $(\{A,B,D,M\} \cup \{B,C,D,E,F,H,O,N\}) mod \{C,F\} = \{A,B,D,E,H,M,N,O\}$ That this has something to do with: $\{A,B,D,E,H,M,N,O\} = (\{A,B,D,M\} \oplus \{B,C,D,E,F,H,O,N\})\, mod \, T$ where $T= \{(-\alpha(x),i_1(x)) \in (\{A,B,D,M\} \oplus \{B,C,D,E,F,H,O,N\}) \forall x \in \{B,C,D,F \}$. I don't see how this relates to the notation such, especially $-\alpha(x)$, what are they trying to say?

2. While I see that $\{A,B,D,E,H,M,N,O\} = \{A,B,D,M\} \oplus \{E,H,O,N\}$ and $\{A,B,D,E,H,M,N,O\}$ is an extension of $\{E,H,O,N\}$ by $\{A,B,D,M\}$ How would I prove this in a general case?

   3. if $\{A,B,D,E,H,M,N,O\}$ is the pushout of $i_1$ and $\alpha$ then is $\{E,H,O,N\}$ the pushout of $i_2$ and $\eta$?

Thanks,

Brian