Q:

An adjective to describe a set, in which each element is different

What would be the idiomatic adjective to describe a set, in which each element is different, for example:
"apple, orange, plum, pear, lemon, peach"
as opposed to a set, in which some elements may be same, for example:
"apple, orange, apple, plum, pear, lemon, plum, peach, lemon, lemon"
?
If possible, please, also provide an adjective for the second set.

A:

Mathematically the first is a "set" the second is a "multiset"  These are technical terms not used outside of maths (and "multiset is not often used in maths) They are nouns, not adjectives.
In casual speech, the distinction is not needed.  It is impossible to have two of exactly the same object (as these two objects would have to be identical in all respects, including position)
So there is no general-purpose adjective to distinguish between these things. In maths there might be a need, and there is a technical term. In general English you'd have to say "a set in which elements can be repeated".

