I'm trying to learn to take a dual of a simplex by understanding the first page of this paper. Here it defines an Alexander dual of a simplicial complex as the set of subsets of vertex set such that the quotient set of vertex set wrt its subset does not belong to the simplicial complex. For a simplicial complex X and a vertex set V, the dual simplicial complex is :
X* = {$\sigma$ $\subseteq$ V | V \ $\sigma$ $\notin$ X}
On the first page of the paper, the example of a simplicial complex S = {{1}, {2}, {3}, {4}, {1,2}, {2,3}, {1,3}, {1,4}, {1,2,3}} and its dual S* = {{1}, {2}, {3}, {4}, {1,2}, {1,3}}is given. I don't get why this is so because for {2,3} $\in$ S, V \ {2,3} = {1, [2], 4} $\notin$ S. So shouldn't {2,3} also be contained in S*?
Even if we assume S = {{1}, {2}, {3}, {4}, {1,2}, {2,3}, {1,3}, {1,4}}, the dual picture does not make sense. Any help with the computation or intuition into the dual would be greatly appreciated.