Consider the system 
$$ x \to \fbox{F} \to y \mbox{.}$$

If we multiply the input signal by $0$, we should have a $0$ output:
$$ 0 \cdot x \to \fbox{F} \to 0 \cdot y $$

This property known as \emph{ZIZO}, can be stated as follows:

$$  \mbox{system is linear} \Rightarrow \mbox{zizo}$$

ZIZO simply means \emph{zero input zero output}. Consider the system defined by: 

%\begin{center}
%\includegraphics{images/modulators/nonzizo.ps}
%\end{center}

\begin{center}
\begin{pspicture}(-3,-3)(4,1.2)

% in from x
\rput(-3.2,0){$x(t)$}
\psline[linewidth=1.25 pt, arrowscale=1.25]{->}(-2.7,0)(-0.25,0)


% out to y
\rput(3.9,0){$y(t)$}

\psline[linewidth=1.25 pt, arrowscale=1.25]{->}(0.25,0)(1.25,0)
% extended
\psline[linewidth=1.25 pt, arrowscale=1.25]{->}(1.75,0)(3.5,0)

% up arrow
\rput(1.5,-2){$g(t)$}
\psline[linewidth=1.25 pt, arrowscale=1.25]{->}(1.5,-1.65)(1.5,-0.25)

% up arrow
\rput(0,-2){$f(t)$}
\psline[linewidth=1.25 pt, arrowscale=1.25]{->}(0,-1.65)(0,-0.25)

% multiplier
\psline{-}(-0.1315,-0.2125)(0.1315,0.2125)
\psline{-}(-0.1315,0.2125)(0.1315,-0.2125)
\pscircle(0,0){0.25}

% adder
\psline{-}(1.25,0)(1.75,0)
\psline{-}(1.5,0.25)(1.5,-0.25)
\pscircle(1.5,0){0.25}


% box
\pspolygon(-1.25,-2.65)(-1.25,1)(2.65,1)(2.65,-2.65)

\end{pspicture}
\end{center}

$y(t) = x(t)f(t) + g(t)$, since zero input still gives us $g(t)$, the system is not linear, since the contrapositive is also true:
$$ \lnot \mbox{zizo} \Rightarrow \lnot \mbox{system is linear} $$

Although some systems may not be linear, they may be \emph{incrementally linear}, for example, consider the following two signals given by
$y_1(t)=f(t)x_1(t)+g(t)$, and $y_2(t)=f(t)x_2(t)+g(t)$. Then $y_1(t) - y_2(t)=f(t)\left[x_1(t)-x_2(t)\right]$ is incrementally linear, or \emph{affine}, meaning that the difference between the two functions is linear.
