Let $M,N$be linear subspaces $L$ then how can we prove that the following map $$(M+N)/N\to M/M\cap N$$ defined by $$m+n+N\mapsto m+M\cap N$$ is surjective? Originally, I need to prove that this map is bijection but I have already proven that this map is injective and well defined,but having hard time to prove surjectivity,please help.
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Given an arbitrary element $x=m+M\cap N$ of $M/(M\cap N)$, note that $m+N\in (M+N)/N$ is mapped to $x$.
Define $T: M \to (M+N)/N$ by $m \mapsto m+N$. Show that it is linear and onto. Check the $\ker T$ and is $M\cap N$ by the first isomorphism theorem $f:M/(M\cap N) \to (M+N)/N$ is an isomorphism.