Method of distinguished element

In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.

Let ${\displaystyle {\mathcal {A}}}$ be a family of subsets of the set ${\displaystyle A}$ and let ${\displaystyle x\in A}$ be a distinguished element of set ${\displaystyle A}$. Then suppose there is a predicate ${\displaystyle P(X,x)}$ that relates a subset ${\displaystyle X\subseteq A}$ to ${\displaystyle x}$. Denote ${\displaystyle {\mathcal {A}}(x)}$ to be the set of subsets ${\displaystyle X}$ from ${\displaystyle {\mathcal {A}}}$ for which ${\displaystyle P(X,x)}$ is true and ${\displaystyle {\mathcal {A}}-x}$ to be the set of subsets ${\displaystyle X}$ from ${\displaystyle {\mathcal {A}}}$ for which ${\displaystyle P(X,x)}$ is false, Then ${\displaystyle {\mathcal {A}}(x)}$ and ${\displaystyle {\mathcal {A}}-x}$ are disjoint sets, so by the method of summation, the cardinalities are additive

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