Cardinality of Subset Relation on Power Set of Finite Set
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Theorem
Let $S$ be a set such that:
- $\card S = n$
where $\card S$ denotes the cardinality of $S$.
From Subset Relation on Power Set is Partial Ordering we have that $\struct {\powerset S, \subseteq}$ is an ordered set.
The cardinality of $\subseteq$ as a relation is $3^n$.
Proof
Let $X \in \powerset S$.
Since $X \subseteq S$, it follows that:
- $X' \subseteq X \implies X' \in \powerset S$
because the Subset Relation is Transitive.
From Cardinality of Power Set of Finite Set, it follows that for any $X \in \powerset S$:
- $\set {X' \in \powerset S: X' \subseteq X}$
has $2^{\card X}$ elements.
Therefore, the cardinality of $\subseteq$ is given by:
- $\ds \sum_{X \mathop \subseteq S} 2^{\card X}$
Let us split the sum over $\card X$:
- $\ds \sum_{X \mathop \subseteq S} 2^{\card X} = \sum_{k \mathop = 0}^n \sum_{\substack {X \mathop \subseteq S \\ \card X \mathop = n}} 2^{\card X}$
It now follows from Cardinality of Set of Subsets that:
- $\ds \card \subseteq = \sum_{k \mathop = 0}^n \binom n k 2^k$
From the Binomial Theorem:
- $\ds \sum_{k \mathop = 0}^n \binom n k 2^k = \paren {1 + 2}^n$
Hence:
- $\card \subseteq = 3^n$
$\blacksquare$
Sources
- 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous): $\S 1$: Sets and Set Inclusion