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 1 of 5: Define translated sets

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A^*=\{x-a:x\in A\},\qquad B^*=\{y+b:y\in B\}.

Detailed analysis

The shifts are chosen so that membership of i+a in A becomes membership of i in A, and membership of i-b in B becomes membership of i in B.