Chapter 10.6, Problem 1TFR

For Questions (a)-(d), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) The unit step function is defined by

$\{\begin{array}{l}0,0\le t\le a\\ 1,t\le a\end{array}$

where $a$ is any positive number.

(b) If $a<b$, the function $f\left(t\right)=u_a\left(t\right)-u_b\left(t\right)$ has value $1$ on the interval $\left[a,b\right]$ and value $0$ elsewhere.

(c) If $a$ and $b$ are positive integers with $a<b$, then $u_a\left(t\right)\le u_b\left(t\right)$ for all $t\ge 0$.

(d) The function

$\{\begin{array}{l}0,0\le t\le a\\ 1,a\le t\le b\\ 0,t\ge b\end{array}$

can be expressed as $f\left(t\right)=u_a\left(t\right)-u_b\left(t\right)$.