MathLabs

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 5 of 5: Cancel the common sums
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∑x∈Ax+∑x∈Bx=∑x∈Ax−a∣A∣+∑x∈Bx+b∣B∣⟹a∣A∣=b∣B∣.\sum_{x\in A}x+\sum_{x\in B}x=\sum_{x\in A}x-a|A|+\sum_{x\in B}x+b|B|\Longrightarrow a|A|=b|B|.
Detailed analysis

Cancel the original sums from both sides of the equality. The remaining balance is exactly a|A|=b|B|.