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← Opštinsko takmičenje (Municipal round) 2021City

Logic and sets · Set operations · Symmetric difference · Periodicity

Problem 2, 2021

CityProof

Let \(A\) and \(B\) be non-empty sets, neither of which is a subset of the other. For a natural number \(n\) consider the equality

\[ \underbrace{A \setminus \bigl(B \setminus (A \setminus (B \setminus \cdots))\bigr)}_{n \text{ sets}} \;=\; \underbrace{A \mathbin{\triangle} \bigl(B \mathbin{\triangle} (A \mathbin{\triangle} (B \mathbin{\triangle} \cdots))\bigr)}_{n \text{ sets}} \]

where on each side the sets \(A\) and \(B\) alternate, the outermost one is \(A\), and \(n\) sets occur in total.

(a) Decide whether this equality holds for \(n = 5\).

(b) Determine all natural numbers \(n\) for which it holds.

Here \(X \mathbin{\triangle} Y\) denotes the symmetric difference of \(X\) and \(Y\), that is, \(X \mathbin{\triangle} Y = (X \setminus Y) \cup (Y \setminus X)\).

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Serbian Municipal Competition 2021, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source

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