Here is the question I am trying to solve:
Characterize paving matroids in terms of their collections of independent sets and in terms of their collection of bases.
What exactly does it mean to characterize paving matroids in terms of their bases? could someone explain this to me please? what exactly should I do to solve this problem?
The usual definition of a paving matroid is a matroid $M$ in which every circuit has size at least as large as $r=\text{rank}(M)$.
Note that this has the form
Therefore, the usual definition is probably* a "characterization in terms of circuits". The question you're being asked is to fill in the following blanks:
and then prove they're true, that is, the same as the circuit characterization. Of course, these statements should not directly reference the circuits.
There could of course be multiple such theorems, but the natural plan of attack is to first recall how you define bases/independent sets in terms of circuits (probably your source has some theorems that prove the equivalence of the various axiom systems; should be around there or in the proofs). Then use that to try to "translate" the fact about circuits ("every circuit has size $\geq r$") to eliminate the circuits.
* It seems like you are being asked this question by someone but you did not understand it. In this case, the right thing to do is to ask for clarification. For instance, if you are taking a class and your professor meant something else, they will probably mark points off... and it is certainly no defense to say "But this random person on the internet said you meant something else!" If you are not taking a course but are doing independent reading, then it may be more productive to email the author; most academics have a public email.