Vector space/Residue class space/Intersection of image spaces/Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}U_{1},U_{2},U$ denote linear subspaces. Let

$\varphi \colon V\longrightarrow V/U$

be the canonical projection. Show that the following statements are equivalent.

a) For the image spaces, we have

${}\varphi (U_{1})\cap \varphi (U_{2})=0\,.$

b) We have

${}U_{1}\cap (U_{2}+U)\subseteq U\,.$

c) We have

${}(U_{1}+U)\cap (U_{2}+U)\subseteq U\,.$

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