I have to answer the following questions:
(a)In a floating-point system, is the unit roundoff $\epsilon_{mach}$ necessarily a machine number? (Explain your answer or give a counterexample).
(b) Is it possible to have a floating-point system in which $\epsilon_{mach} \lt UFL$ (underflow level)? If so give an example
My attempt:
(b) Let $\{\beta, p, L, U\}$ a floating-point system. We know that $\epsilon_{mach} = \beta ^{1-p}$ and $UFL = \beta ^ L$. We should choose $p$ and $L$ such that $\beta ^ {1-p} \lt \beta ^ L$, i.e., $-L \lt p-1$ in order to Have the desidered propriety. For example $p=3$ and $L=-1$, in this case, $\epsilon_{mach} = \beta^{-2} \lt\beta^{-1}=UFL$
(a) The only thing that comes to my mind is to choose a floating-point system such that $\epsilon_{mach} \lt UFL$ just as in (b). Is there another example when $\epsilon_{mach}$ is not a machine number?.
And I have another question: Is it necessary $L$ to be greater than zero?
As you pointed out, the unit roundoff of a floating-point system is based only one the base $\beta$ and the precision $p$, while the underflow level is based on the base $\beta$ and the smallest exponent of the system $L$ i.e. $\varepsilon_{\text{mach}} = \beta^{1-p}$, $\text{UFL} = \beta^{L}$.
From this follows that the roundoff is not necessary a machine number, it is only when $1-L \le p$ and you can build a system satisfying (b) as you shown.
L is not necessary to be greater then zero, for example in the IEEE-754 is negative (e.g. Double precision format has $(\beta, p, L, U) = (2,53,-1022,1023)$).