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:

$\displaystyle \sum_{X \mathop \subseteq S} 2^{\card X}$

Let us split the sum over $\card X$:

$\displaystyle \sum_{X \mathop \subseteq S} 2^{\card X} = \sum_{k \mathop = 0}^n \sum_{\substack {X \mathop \subseteq S \\ \card X \mathop = n}} 2^{\card X}$

It now follows from Cardinality of Set of Subsets that:

$\displaystyle \card \subseteq = \sum_{k \mathop = 0}^n \binom n k 2^k$


From the Binomial Theorem:

$\displaystyle \sum_{k \mathop = 0}^n \binom n k 2^k = \paren {1 + 2}^n$

Hence:

$\card \subseteq = 3^n$

$\blacksquare$


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