Problem 4
Let a and b be positive integers, and let A and B be finite sets of integers satisfying: (i) A and B are disjoint; (ii) if an integer i belongs either to A or to B, then i+a belongs to A or i-b belongs to B. Prove that a|A|=b|B|, where |X| denotes the number of elements of X.
Step 3 of 5: Use the equality case
Detailed analysis
The first equality follows from the inclusion and equal finite cardinalities. Equality in the union cardinality bound says that A and B are disjoint.