Which of the following are partitions of A={a, b, c, d, e,...

Which of the following are partitions of $A=\{a, b, c, d, e, f, g\}$ ? For each collection of subsets that is not a partition of $A$, explain your answer. (a) $S_1=\{\{a, c, e, g\},\{b, f\},\{d\}\}$ (b) $S_2=\{\{a, b, c, d\},\{e, f\}\}$ (c) $S_3=\{A\}$ (d) $S_4=\{\{a\}, \emptyset,\{b, c, d\},\{e, f, g\}\}$ (e) $S_5=\{\{a, c, d\},\{b, g\},\{e\},\{b, f\}\}$.

Which of the following are partitions of $A=\{a, b, c, d, e, f, g\}$ ? For each collection of subsets that is not a partition of $A$, explain your answer.
(a) $S_1=\{\{a, c, e, g\},\{b, f\},\{d\}\}$
(b) $S_2=\{\{a, b, c, d\},\{e, f\}\}$
(c) $S_3=\{A\}$
(d) $S_4=\{\{a\}, \emptyset,\{b, c, d\},\{e, f, g\}\}$
(e) $S_5=\{\{a, c, d\},\{b, g\},\{e\},\{b, f\}\}$.

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Key Concepts

Non-emptiness of Subsets

Each subset in a partition must be non-empty. The inclusion of the empty set disrupts the idea of a partition as it would represent a ‘missing block’ rather than a defined portion of the set.

Complete Covering (Union Equals the Set)

A valid partition must cover the entire set, meaning that the union of all the subsets must yield the original set. This ensures that no element is left out and every element is assigned to one and only one subset.

Partition of a Set

A partition of a set is a collection of nonempty subsets that are mutually exclusive (i.e. pairwise disjoint) and whose union equals the entire original set. This concept is fundamental in set theory because it breaks down a set into ‘blocks’ that do not overlap and together cover all elements.

Pairwise Disjointness

In a partition, every two distinct subsets must have no element in common. This property guarantees that each element of the original set belongs to exactly one subset, eliminating ambiguity in element allocation.

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