In your example it is possible because the space is finite (so we actually have an algebra instead of a sigma-algebra).
We just need a procedure which starts with the family C and takes finite unions of finite intersections of the sets in C and their complements,  until it stabilizes. 

Unfortunately such a proc is not very useful because the resulting (sigma)algebra is in general HUGE.

BTW, for your example X := {1, 2, 3}, C := {{1}, {2}},  the algebra generated by C is obviously the powerset P(X) of X (i.e. it is maximal, containing 2^3 sets). 
 

@Kitonum Actually, the answer is NO. For example, taking f := alpha the integral R-S does not exist but Maple happily "computes" it!

@Kitonum Of course. And this does not work for subexpressions e.g.  expand(f(tan(x+k*Pi))) ... 

@Kitonum It's sad that Maple cannot simplify:
simplify(tan(x+k*Pi))    assuming   k::integer;
simplify(sin(x+2*k*Pi)) assuming   k::integer;
 

So, you have two m x n matrices A and B.
You need a (column) permutation matrix P and a (row) permutation matrix Q  such that Q.A.P = B, if such P, Q exist.
(actually, it seems that you are interested only in P, and for Q the existence is enough).
Is this correct? Do you need all the possibilities for P?
 

@JAMET Then, do not assign X,Y:

[X = (a*m^2 + 2*m^2*p + 2*p)/(2*m^2), Y = a/(2*m)]:
eliminate(%, m);

(a parabola)

Actually, in modern mathematics these symbols are considered redundant.

You've got four answers and no reaction. It's not a polite attitude!

@Earl The method is mentioned in the help page for EulerLagrange (for the case of a single function). They are not important, but they could simplify the computations sometimes.

In Windows both examples work fine.

@mmcdara The Riemann sphere is defined via the stereographic projection, see https://encyclopediaofmath.org/wiki/Riemann_sphere

It does not introduce singularities. Unfortunately it will not help  the OP in better understanding complex functions.

Int(cos(x)/(x^2+1),x=-infinity..+infinity) = 2*Pi*I*residue(exp(I*x)/(x^2+1), x=I);

        

@janhardo Edwards's book contains several obsolete notations including some used originally by Riemann himself. It is not a good idea to try to learn Complex Analysis from such a source (even if the book is a valuable monograph).   

You are a beginner in Complex Analysis. So, you should start with standard exercises. In this one, the path must be exactly defined; probably here we actually have a limit of a complex integral (e.g. tha radius of the arc around the origin tends to 0). 
The branches could be a problem; maybe it is not the Maple principal branch.
Not to mention z; the integral could diverge for some z, depending on how the path is defined.

dx / x is just for a convenient way of writing the integral: ∫ f(x)/x dx =  ∫ f(x) dx/x.

BTW, the notation Int( ..., x = +oo .. +oo)  is very exotic. Where did you find it?

@John2020 The methods do not work when the parameters depend on t; actually in such a case the problem does not make much sense.