Bdtyktl

I am reading about the Satisfiability Problem, in page (5) the author gives the following example :

$(P lor Q lor R) wedge
(bar{P} lor Q lor bar{R}) wedge
(P lor bar{Q} lor S) wedge
(bar{P} lor bar{R} lor bar{S})$

A satisfying assignement is $: P,Q,bar{R},bar{S}$.

A different assignments is $P,bar{R}$ to satisfy the four caluses. This means that $Q,S$ can be set arbitrarily to $True$ or $False$.

Do we require an assignment for all variables or we only reqiure to set variables that satisfy the input (i.e. $P,bar{R}$ discluding $Q,S$) ?

Does that mean that the satisfying assignment of $P,bar{R}$ is more efficient than $P,Q,bar{R},bar{S}$ given that it uses less variables ? Is there any resources to read about it ?

Also, given that $P,bar{R}$ satisfy the input, does that mean that we can disclude $Q,S$ from the original Boolean Formula ?

If the truth value of a formula is determined by setting only a subset of the variables, an author might skip describing the remaining truth values. However, by definition, a truth assignment gives a value to every variable.