Question of maps in Mayer-Vietoris sequence We obtain MV-seq. from short exact sequence
$$
0\to C_n(A\cap B) \to C_n(A)\oplus C_n(B)\to C_n(A+B)\to 0
$$
So map i wonder that map $H_n(A\cap B)\to H_n(A)\oplus H_n(B)$ maps $[a]$ to $([a],[-a])$. But for example in this case it's not quite clear why $1 \mapsto (2,-2)$. Why $1 \not\mapsto (1,-1)$? How to make argument 'wraps twice' rigorous?
And what about other maps in this sequence?

You have to compute the homomorphisms $H_*(\partial M) \to H_*(M)$, induced by inclusion, where $M$ is the Mobius band.  How to do this depends on what you know/understand about homology.  Do you know what a deformation retraction does to homology groups?  
If you know about cellular homology, you can compute $C^{\rm cell}_*(\partial M) \to C^{\rm cell}_*(M)$, and then see the maps on homology very directly. 
