Differences of between ''either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$'' AND ''for all $x$ either $f(x)=0$ or $g(x)=0$''

Question

• Let $f,g$ be two functions from $\mathbb{R}$ to $\mathbb{R}$. Assume for all $x$ $f(x)g(x)=0$, then

i) then either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$.

ii) then for all $x$ either $f(x)=0$ or $g(x)=0$.

My answer is:Assume for all $x$ the equation $f(x)g(x)=0$ holds. Let $f\left( x\right) =\begin{cases}1, & x=1\\ 0,& x\neq 1\end{cases}$ and $g\left( x\right) =\begin{cases}1, & x=2\\ 0,& x\neq 2\end{cases}$. So, these works for the equation but it doesn't imply that for all $x$, $f(x)$ or for all $x$, $g(x)$ must be zero.

Can you check that does my answer works i) and ii)? And can you explain Differences of between ''either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$'' AND ''for all $x$ either $f(x)=0$ or $g(x)=0$''?

Answer 1

When you say either $f(x)=0$ for all $x$ or $g(x)=0$ for all $x$ you have to have one of your functions identically zero for all $ x$.

On the other hand for all $x$ either $f(x)=0$ or $g(x)=0$ simply means the product of your functions must be identically zero.

As you see, the first statement implies the second one but the second does not imply the first one.

Answer 2

What is the difference between :

"either $P(x)$ for all $x$ or $Q(x)$ for all $x$" AND "for all $x$ either $P(x)$ or $Q(x)$".

Consider :

"either all (natural) numbers are even or all (natural) numbers are odd" versus "all (natural) numbers are either even or odd".