- #1

Devil Moo

- 44

- 1

1. Homework Statement

1. Homework Statement

Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other".

## The Attempt at a Solution

Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W## is a subspace of ##V##. (statement A)

Suppose ##U## is not contained in ##W## and vice versa. (statement B)

Let ##u \in U, \not\in W## and ##w \in W, \not\in U##.

##u \in U \cup W, w \in U \cup W##

##u + w \in U \cup W##

##u + w \in U## or ##u + w \in W##

Suppose ##u + w \in U## (statement C)

##u + w + (-u) \in U##

##w \in U##

It leads to contradiction.

Which one does it conclude? (if A is true, then B is false) or (if A and B are true, then C is false)