For a family $A$ of subsets of a ground set $[n]=\{1,\ldots,n\}$, its filter is the family $A^{\triangledown}$ consisting of all subsets containing at least one member of $A$. Let (as in the book) $b(A)$ denote the family of minimal (under under set-inclusion) subsets of $[n]$ that intersect all members of $A$.
Lemma: $|A^{\triangledown}|+|b(A)^{\triangledown}|=2^n$.Proof: It is enough to show that for every subset $s\subseteq [n]$, $ s\in A^{\triangledown} \iff \overline{s}\not\in b(A)^{\triangledown}. $ By the definition of $b(A)$ and of the filter, we have that $\overline{s}\not\in b(A)^{\triangledown}$ iff $\overline{s}\cap a=\emptyset$ for some $a\in A$ iff $s\supseteq a$ for some $a\in A$ iff $s\in A^{\triangledown}$. $\Box$
Recall that a family $A$ is called self-dual if $b(A)=A$.
Corollary: An antichain $A$ is self-dual if and only if $|A^{\triangledown}|=2^{n-1}$.For example, the families $A=\{\{1\}\}$ and $B=\{\{1,2\},\{1,3\},\{2,3\}\}$ on the ground set $[3]=\{1,2,3\}$ have no structural similarity but both are self-dual because $|A^{\triangledown}|=|\{\{1\}, \{1,2\},\{1,3\},\{1,2,3\}\}| =4=2^{3-1}$ and $|B^{\triangledown}|=|\{\{1,2\},\{1,3\},\{2,3\}, \{1,2,3\}\}|=2^{3-1}$.
The corollary has an interesting implication for boolean functions $f:\{0,1\}^n\to\{0,1\}$. By identifying a binary vector $(a_1,\ldots,a_n)\in\{0,1\}^n$ with the set $a=\{i\colon a_i=1\}$ of its $1$-positions, we can view each such function as a $2$-coloring $f:2^{[n]}\to\{0,1\}$ of the family $2^{[n]}$ of all $2^n$ subsets of $[n]$; here $f(a)=1$ means that the function "accepts" the set $a$, and $f(a)=0$ that it "rejects" $a$. A boolean function $f$ is
The corollary implies that a monotone boolean function is self-dual if and only if it accepts exactly half of the input sets.
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