If we are given A,B,C be 3 subspaces of a vector space then which of the following id correct.
a)- $A \cap( B + C ) = (A + B) \cap (A + C)$
b)- $A \cap( B + C )\subset (A + B) \cap (A + C)$
c)- $ (A + B) \cap (A + C)\subset A \cap( B + C ) $
If $A=\{0\} $
then $ A \cap( B + C )=\{0\}$ but
$(A + B) \cap (A + C)= B \cap C$
Hence $A \cap( B + C ) \neq (A + B) \cap (A + C)$
Infact we also have $(A + B) \cap (A + C)\not\subset A \cap( B + C )$
Hence a) and c) are False(IS IT CORRECT)
What about b)? I can't figure out about b). Help Me
Option (b) is true. Suppose $ p \in A \cap (B+C) $ then $p \in A $ i.e. $p \in A+ B$ since p is in A and 0 is in B so p = p + 0 is in A + B . Similarly p is in A + C . So $p \in (A + B) \cap (A + C) $ .