The question is bellow can you explain in detail the solution.

83085 assignment
First let us understand the important definitions for this proof.
A set is countable if it is finite or countably infinite
A set is finite if it contains a limited number of elements (thus it is possible to list every single element in the set)
A set is countably infinite if the set contains an unlimited number of elements and if there is one-to-one correspondence with the positive integers.
List of all integers between -5 and +5 are -5,-4,…+5. i.e. there are 10 in number.
The quadratic equation given can have at most two roots.
A,b,c can be selected with repition in 10^3 possible ways at the most. (Note: a cannot be 0, so for quadratic there are 900 ways possible)
The roots would be at maximum 900*2 = 1800 hence finite.
So cardinality can be at the most 1800 and it may be less if repetition of roots is there.
K is a positive integer and S(k) is the set of all real...