[Math] For arbitrary subspaces U,V and W of a finite dimensional vectorspace , which of the following relations hold

Question

For arbitrary subspaces U,V and W of a finite dimensional vectorspace , which of the following hold

a)U$\cap$(V+ W) $\subset$ U$\cap$V + $U\cap W $

b)U$\cap$(V+ W) $\supset$ U$\cap$V + $U\cap W $

c)(U$\cap$V)+ W $\subset$ (U +W) $\cap$ (V+W)

d)(U$\cap$V)+ W $\supset$ (U +W) $\cap$ (V+W)

I don't know to solve this.This is a CSIR NET QUESTION and they have given answer as options b) and c).Please help…

Answer 1

Hint- use the fact that if $W_{1}$ and $W_{2}$ are two subspace of a vector space $V$ then $W_{1}\cap W_{2}$ is the largest subspace of $V$ contained in $W_{1}$ and $W_{2}$ and $W_{1}+W_{2}$ is the smallest subspace containing both $W_{1}$ and $W_{2}$.