I've edited this question to make it more understandable:
See the following pairs of descriptions of sets.
'n mod 2 = 0', '2 * int(n)'
'sqrt(x^2+y^2) = 1', 'x = sin(n) y = cos(n)'
'?', 'Collatz algorithm starting from 27, returning the nth element'
The left of each pair can be queried for if a given number is part of the set, from the first example '4 mod 2 = 0' and the answer in this case is yes. This is usefull but it may take forever to find even one number from a set with only a few elements.
The right part represents the same set but can be queried for a number from the set in no particular order, so for example to get 5 numbers from the set is easy and fast but one can't ask if a given number is part of the set easily if at all for a set of infinite size.
Can I convert between thease two forms, if so then how?
old form of the question
'n mod 2 = 0' can be plotted to show a row of equally spaced peeks (where peak is a word I use to mean a value of true sorrounded by values of false), you can ask if a given position is a peek by replacing n with the value you want like this '2 mod 2 = 0', in this case yes, there is a peak at position 2. The equation can presumably be rearranged to get the position of the n'th peek, for example 2n would work.
These two different approaches can apparently be referred to as push and pull methods of getting the information.
Is there a way to convert between these two ideas?
'n mod 2 = 0' <-> '2n'
A less trivial example:
x = sin(n) y = cos(n) Wolfram Alpha
sqrt(x^2+y^2) = 1 Wolfram Alpha
Where the first version gives every part of the outside of the circle when run for every possible n and the second version gives if a position is on the outside of the circle.
Returning to the first example. Completing all peaks using '2n' to find out if any lay on a given position would take forever, however you may only have an algorithm that works by repeatedly placing a peak and moving forward two units and you may not know about the mod function or that 'n mod 2 = 0' is the reverse of your algorithm.
A more complicated but perhaps meaningfull example is a en.wikipedia.org/wiki/Collatz_conjecture starting from a given number might go on forever, or for a very long time, described algorithmically it is like 'n2' in the first example but if we want to know if a given number is part of the sequence then we need the 'n mod 2 = 0' version of the equation.