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Omni-MATH / Let S={1,2,…,n}, where n≥1. Each of the 2^(n) subsets of S is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set T⊆S, w…

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problem

Let S={1,2,…,n}S = \left\{ 1,2,\dots,n \right\}, where n≥1n \ge 1. Each of the 2n2^n subsets of SS is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set T⊆ST \subseteq S, we then write f(T)f(T) for the number of subsets of TT that are blue. Determine the number of colorings that satisfy the following condition: for any subsets T1T_1 and T2T_2 of SS, f(T1)f(T2)=f(T1∪T2)f(T1∩T2). f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).
Plain-text mathematical notation (without MathML)
Let S={1,2,…,n}, where n≥1. Each of the 2^(n) subsets of S is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set T⊆S, we then write f(T) for the number of subsets of T that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets T₁ and T₂ of S, f(T₁)f(T₂)=f(T₁∪T₂)f(T₁∩T₂).
Original LaTeX notation
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]

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Source and history

task_id
bc54c3d9-3c1c-5161-89f8-69c44ee9b39d
task_key
test--bc54c3d9-3c1c-5161-89f8-69c44ee9b39d
task_revision_id
1
config / split
/ test

Official source

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