Let A={1,2, …, 12}. Give an example of a partition S of A...

Let $A=\{1,2, \ldots, 12\}$. Give an example of a partition $S$ of $A$ satisfying the following requirements: (i) $|S|=5$, (ii) there is a subset $T$ of $S$ such that $|T|=4$ and $\left|\cup_{X \in T} X\right|=10$ and (iii) there is no element $B \in S$ such that $|B|=3$.

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Key Concepts

Exclusion Conditions in Partitions

Sometimes, additional restrictions are imposed to avoid certain configurations—for example, disallowing a block of a specific size. Such exclusion conditions require designing partitions that not only meet numerical and union constraints but also avoid forbidden block sizes, adding an extra level of complexity to the combinatorial construction.

Union of Subsets

The union of subsets gathers all elements from multiple blocks without repetition. Understanding how the size of the union relates to the sizes of individual blocks is key, especially when a problem imposes conditions on the number of distinct elements present when several blocks are combined.

Set Partition

A set partition divides a set into disjoint nonempty subsets (blocks) such that every element of the set is included in exactly one block. This concept is foundational in combinatorics, providing a structured way to break down a set according to specific criteria.

Cardinality Constraints

In many combinatorial problems, specific requirements on the number of elements in subsets or the total number of subsets must be met. This includes specifying the total number of blocks in the partition as well as conditions on the size of the union of some blocks, ensuring the solution satisfies exact element-count constraints.

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