Argthtjtr

Claim. Let $V$ be a vector space over $F$, and suppose that $W_1, W_2,$ and $W_3$ are subspaces
of $V$ such that $W_1 + W_3 = W_2 + W_3.$ Then $W_1 = W_2.$

Proof

$W_1 + W_3 - W_3 = W_2 + W_3 - W_3$

$therefore$ $W_1 = W_2$

My proof feels like cheating, is it even valid? Also is there anyway to 'say', prove or demonstrate the same thing in a more rigorous way i.e. set builder notation?

In regards to the rules of the site is my question to simple, if so what can I do to make it better?

Consider $V=mathbb{R}^2 $ and $W_3= {(x,0) : xin mathbb{R}}$, $W_1= {(0,y) : yin mathbb{R}}$ and $W_2= {(t,t) : tin mathbb{R}}$

You should be able to verify that $W_1+W_3 = W_2 + W_3$

What does that tell you about your claim?

What does your claim say if $W_3$ is the whole space $V$?

You can't subtract subspaces.

There is a very simple counterexample. Since the statement is about arbitrary subspaces $W_1$, $W_2$ and $W_3$ only subject to the condition that $W_1+W_3=W_2+W_3$, it should in particular hold for $W_1={0}$, $W_2=W_3=V$. Then from
$$
{0}+V=V+V
$$
(which is true), you'd conclude that
$$
{0}=V
$$
Now any non trivial vector space is a counterexample.

From a slightly higher point of view, the set $mathscr{L}(V)$ of subspaces of $V$ is a commutative monoid under the $+$ operation, because it is associative and has the neutral element ${0}$, because ${0}+W=W$, for every $Winmathscr{L}(V)$.

However, this monoid cannot be cancellative (that is $W_1+W_3=W_2+W_3$ implies $W_1=W_2$) for several reasons, the most important one being that it has an absorbing element, namely $V$:
$$
W+V=V
$$
for every $Winmathscr{L}(V)$. An absorbing element cannot have a symmetric element, unless the monoid is trivial.

Also, if $U$ is a subspace of $V$ and $mathscr{L}(V;subseteq U)$ denotes the set of subspaces of $V$ contained in $U$, we have $mathscr{L}(V;subseteq U)=mathscr{L}(U)$, which is thus a submonoid of $mathscr{L}(V)$. Since every $Uinmathscr{L}(V)$ is the absorbing element in a submonoid, no element can have a symmetric element, except for ${0}$.

Note. By “symmetric element” of an element $x$ in a monoid $(M,*,e)$ I mean an element $y$ such that $x*y=y*x=e$.