show that G is a sigma-algebra

1. The problem statement, all variables and given/known data

A σ-algebra G on a set X is a family of subsets of X satisfying:



1) X[itex]\in[/itex]G

2)A[itex]\in[/itex]G => C(A)[itex]\in[/itex]G

3)A_j [itex]\subset[/itex] G => [itex]\bigcup[/itex] A_j [itex]\in[/itex] G



Show that G = {A[itex]\subset[/itex]X : #A≤N or ≠C(A)≤N}



# stands for the cardinality of the set.

2. Relevant equations







3. The attempt at a solution

Actually I am not so far in the problem solving because I am stuck at showing the first property. We must have that X[itex]\in[/itex]G. But since G is only the set of proper subsets of X, i.e. doesn't contain X by definition, how can 1) hold?

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