Many students are slightly mystified about what exactly the running time T$($n$)$ as a function of n means, and$$that$$s not exactly surprising. After all, the running time of a program$/$algorithm depends on its entire input, not$$just the size of the input. For some inputs of size n$,$ the algorithm might be fast, and for others, it might be slow.$$So really, the running time can be written as a function Tinput$($I$),$ where I is the input of the algorithm. But we$$usually want to get an idea of how the running time grows as a function of the input$$s size. So we define $$and mostly focus on $$ T$($n$)=$ maxinputs I of size n Tinput$($I$).($This is the definition of running time as a function$$of n$.)$ Now, let$$s explore what this means for running time upper and lower bounds. For both of the following$$sub$-$questions, in your answers, you should not be referencing any $$maximum$$ or similar things. Instead, you$$will want to use $$for all$$ and $$exists$$ clearly, i$.$e$.,$ your text $($or formulas if you prefer$)$ should be describing$$quantifiers$,$ and talking about the running time on inputs.$($a$)$ Suppose that you want to show that T$($n$)=$ O$($f$($n$)).$ What do you need to show?$($b$)$ Suppose that you want to show that T$($n$)=\backslash$Omega $($g$($n$)).$ What do you need to show?$$Both of these parts are not very hard if you are precise in applying the definitions and thinking through them.$$But it$$s easy to get the second one wrong if you just try to $$wing it$.$