A finite family $(A_i{)}_{i\in I}$ of subsets of a finite set $X$, the intersections $A_J$ for $J\subseteq I$, and the convention $A_{\mathrm{\varnothing }}=X$

A sieve family consists of a finite set $X$, called the ambient set, a finite set $I$, called the index set, and a family $(A_i{)}_{i\in I}$ of subsets of $X$, that is a function $I\to \mathcal{P}(X)$ (Finite, countably infinite, countable, uncountable, The cardinality $\mid A\mid$ of a finite set). For $J\subseteq I$ set

$A_J:=\{\begin{array}{ll}{\displaystyle \bigcap _{i\in J}{A}_i,}& J\mathrm{\ne }\mathrm{\varnothing },\\ X,& J=\mathrm{\varnothing },\end{array}$

and write $U:={\bigcup }_{i\in I}{A}_i$ for the union of the family.

Why the ambient set has to be named, and why $A_{\mathrm{\varnothing }}:=X$ is a stipulation. For $J\mathrm{\ne }\mathrm{\varnothing }$ the intersection ${\bigcap }_{i\in J}{A}_i$ is the set of elements belonging to every $A_i$ with $i\in J$, and it is determined by the family alone. For $J=\mathrm{\varnothing }$ that description is satisfied by every set whatsoever, so it determines nothing; an intersection of no subsets of $X$ is $X$ only relative to $X$. Naming $X$ as part of the data and stipulating $A_{\mathrm{\varnothing }}=X$ is what makes the symbol $A_J$ defined for all $J\subseteq I$, which is what the complementary form of the sieve identity requires.

(a) Every $A_J$ is a finite subset of $X$. For $J\mathrm{\ne }\mathrm{\varnothing }$ pick $i\in J$; then $A_J\subseteq A_i\subseteq X$. For $J=\mathrm{\varnothing }$, $A_J=X$. In both cases $A_J\subseteq X$ is finite by clause 1 of A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$, and so is $U\subseteq X$; hence $\mid A_J\mid$ and $\mid U\mid$ are natural numbers (The cardinality $\mid A\mid$ of a finite set).

(b) The index sets of the sieve's sums are finite. $\mathcal{P}(I)$ is finite with $\mid \mathcal{P}(I)\mid =2^{\mid I\mid }$ ($\mid \mathcal{P}(A)\mid =2^{\mid A\mid }$ for finite $A$); the set $\mathcal{P}(I)\setminus \{\mathrm{\varnothing }\}$ of nonempty subsets of $I$ and the set $[I{]}^j$ of $j$-element subsets of $I$ are subsets of $\mathcal{P}(I)$, hence finite (A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$), and $\mid [I{]}^j\mid =\left(\genfrac{}{}{0px}{}{\mid I\mid }{j}\right)$ (The set $[A{]}^k$ of $k$-element subsets and the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{k}\right):=\mid [n{]}^k\mid$).

(c) Monotonicity. If $J\subseteq J^{\mathrm{\prime }}\subseteq I$ then $A_{J^{\mathrm{\prime }}}\subseteq A_J$. For $J=\mathrm{\varnothing }$ this is clause (a); otherwise an element lying in every $A_i$ with $i\in J^{\mathrm{\prime }}$ lies in every $A_i$ with $i\in J$.

(d) The trace of a point. For $x\in X$ put

$T(x):=\{i\in I:x\in A_i\}\subseteq I,$

a finite set. For every nonempty $J\subseteq I$,

$x\in A_J⟺J\subseteq T(x),$

both sides saying that $x\in A_i$ for every $i\in J$. And $x\in U$ if and only if $T(x)\mathrm{\ne }\mathrm{\varnothing }$. Writing $t(x):=\mid T(x)\mid$, clause (b) applied to $T(x)$ gives $\mid [T(x){]}^j\mid =\left(\genfrac{}{}{0px}{}{t(x)}{j}\right)$, and for nonempty $J$ the condition $x\in A_J$ with $\mid J\mid =j$ says exactly that $J\in [T(x){]}^j$.

• The counts stay in $\mathbb{N}$; the identities do not. Each $\mid A_J\mid$ is a natural number. Every identity that sieves them carries a minus sign, and $\mathbb{N}$ has no subtraction, so those identities are stated in $\mathbb{R}$ through the canonical natural and read back by its injectivity. That is a property of the identities, not of this definition, which introduces no arithmetic at all.

• $I$ is an arbitrary finite index set, not a natural number. Nothing below numbers the sets $A_0,A_1,\dots$; the subsets $J\subseteq I$ are the objects the sums run over, and $\mid J\mid$ rather than any position is what carries the sign.

• The clause $A_{\mathrm{\varnothing }}=X$ supplies every empty-subfamily term. In the complementary form at $J=\mathrm{\varnothing }$ it contributes $\mid X\mid$; later sieve instances also use it when identifying the intersection at the empty subfamily. Removing the stipulation would leave those terms undefined.