Hw2 Problem 12

Problem 12

Let $F$ be the set of all real-valued functions having as domain $\mathbb R$. Prove or give a counterexample for the following statements.

(a) Function addition $+$ on $F$ is associative.
(b) Function subtraction $-$ on $F$ is commutative.

Solution

(a) Function addition $+$ on $F$ is associative.

Since addition is associative on $\mathbb R$:

(1)

\begin{align} [(f+g)+h](x)\\ =(f+g)(x) + h(x)\\ =f(x) + g(x) + h(x)\\ =f(x) + (g+h)(x)\\ =[f+(g+h)](x)\\ \end{align}

Yes, $+$ on $F$ is associative.

(b) Function subtraction $-$ on $F$ is commutative.
For $f(x) = 2x$ and $g(x) = x$,

(2)

\begin{align} 2x-x=x\\ x-2x=-x\\ x \neq -x \end{align}

No, $-$ on $F$ is not commutative.