# Math Help - Closure of sets

1. ## Closure of sets

A function f with domain $(0,\infty)$ and co-domain $R$ is log-like if
$f(x+y)=f(x)+f(y)$

Show the set of all log-like functions is closed under addition

Attempted Proof
Let functions f and g be log-like
Consider f+g

$f(xy)+g(xy)=f(x)+f(y)+g(x)+g(y)
$

$(f+g)(xy)=(f+g)(x)+(f+g)(y)$

Thus this set is closed
Is this operation correct?

2. The top equation should be $f(xy) = f(x) + f(y)$. Then your proof makes sense since $(f+g)(x) = f(x) + g(x)$

3. Actually your proof is a bit unclear. I think the following is better (using what Haven said):

$(f+g)(xy)=f(xy)+g(xy) = f(x)+f(y)+g(x)+g(y)=f(x)+g(x)+f(y)+g(y)=(f+g)(x)+( f+g)(y)$