The conventions this page fixes: the empty intersection, where the counts live, the first index of every sum, and what the declared prerequisites do not supply

This item is the page's ledger: every convention the page fixes, with the item that fixes it, and a statement of what this page's declared prerequisites do not supply. It continues Conventions fixed on this page, and what counting is deliberately not done here, the same kind of ledger for the page finite-counting-and-binomial-coefficients, which is this page's declared prerequisite.

The empty intersection is the ambient set, and the ambient set is part of the data. 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 $A_\varnothing = X$ fixes a finite $X$ together with the family $(A_i)_{i \in I}$ and stipulates $A_\varnothing = X$. This is a stipulation and not a theorem: for nonempty $J$ the intersection is determined by the family, and for $J = \varnothing$ the description "belongs to every $A_i$ with $i \in J$" is satisfied by everything, so it determines nothing without an ambient set to be relative to. The clause supplies the $J = \varnothing$ term in the complementary form of Inclusion and exclusion: $\iota\lvert\bigcup_{i \in I} A_i\rvert = \sum_{\varnothing \ne J \subseteq I}(-1)^{\lvert J\rvert + 1}\,\iota\lvert A_J\rvert$, together with the complementary form counting the elements in none of the $A_i$ and in later sieves whenever an intersection must also be identified at the empty subfamily.

Counts live in $\mathbb{N}$; every alternating identity lives in $\mathbb{R}$. A cardinality is a natural number, a natural number here is a set, and a set is not an element of $\mathbb{R}$. Every identity on this page that uses a negative summand, and every one that divides counts, is therefore stated in $\mathbb{R}$ with the counts carried across by the canonical natural $\iota$ (The canonical natural $\iota(n) = n \cdot 1_F$ of a field), and read back through the injectivity of $\iota$ where the conclusion is about natural numbers. Truncated natural differences such as $n-1$ and $n-2$ remain in $\mathbb N$ and are not negative summands.

Every index range starts at $0$, and the lower-bound hypotheses on this page exist only because of it. $\sum_{j<m+1}(-1)^{j}\,\iota\binom{t}{j} = (-1)^{m}\,\iota\binom{t-1}{m}$ for every $t \ge 1$ and every $m$ carries $t \ge 1$; without it the identity fails at $t = 0$ and $m = 1$. $\iota(D_n) = \iota(n)\,\iota(D_{n-1}) + (-1)^{n}$ for $n \ge 1$, and $D_n = (n-1)(D_{n-1} + D_{n-2})$ for $n \ge 2$ carries $n \ge 1$ in its first clause, because the identity $n! = (n-1)!\cdot n$ that proves it fails at $n = 0$ under the truncated difference, and $n \ge 2$ in its second, because that clause applies the first one at $n-1$. $\lceil m/n \rceil$ for naturals $m$ and $n \ge 1$: the least $q \in \mathbb{N}$ with $m \le n q$ carries $n \ge 1$, without which the set it takes a least element of can be empty.

$0^{0} = 1$, and the place it is spent is named. The convention is the base clause of the recursion in Exponentiation of natural numbers, $m^{n}$, and its agreement with the integer power in $\mathbb{R}$, not an import. In The number of surjections from an $n$-element set onto a $k$-element set is $\sum_{i<k+1}(-1)^{i}\binom{k}{i}(k-i)^{n}$, read in $\mathbb{R}$ through $\iota$ it is what makes the formula correct at $n = 0$ and $k = 0$, where the empty function is the unique surjection $\varnothing \to \varnothing$ and the formula returns $\iota(0^{0})$. At $n = 0$ with $k \ge 1$ the same formula returns the full alternating row sum, which vanishes only because $k \ge 1$; and at $k = 0$ with $n \ge 1$ it returns $\iota(0^{n})$, which is $0$ for the same reason read the other way. Powers of $-1$ are the real powers of Integer powers $a^m$ throughout, since $-1$ is not a natural number.

A sum over a finite index set is what all of this is written in. The sum $\sum_{i \in S} a_i$ over a finite index set, and its product form is the notion used for every sum whose index set is a set of subsets, a set of elements or a relation's fibre family, and its value is independent of the enumeration chosen. The facts about it that are used constantly and are not clauses of it are the bridge $\iota\big(\sum^{\mathbb{N}}\big) = \sum^{\mathbb{R}}\iota$ over such an index set, and the additivity, scaling and monotonicity laws over such an index set. Both are derived, in the Facts of the items that use them, from the corresponding clauses about a sum over an initial segment together with the enumeration that defines the sum over an index set.

A relation, not a matrix. A relation $R \subseteq X \times Y$ between finite sets, its row fibres $R_x$ and its column fibres $R^y$ sets up a subset of a product of two finite sets, its two fibre families and their slice partitions. Double counting: $\sum_{x \in X}\lvert R_x\rvert = \lvert R\rvert = \sum_{y \in Y}\lvert R^y\rvert$ for a relation between finite sets states the resulting equality of the two fibre sums with the size of the relation, and nothing more.

Fibres of a function. If $\lvert A\rvert > k\lvert B\rvert$ then every $f : A \to B$ has a fibre with more than $k$ elements, and for nonempty $B$ some fibre has at least $\lceil \lvert A\rvert / \lvert B\rvert\rceil$ elements is stated about the fibres $f^{-1}[\{b\}]$ of a function between finite sets. Its two clauses are one argument: the ceiling form is the counting form applied at the single index below the ceiling, which is why the ceiling was defined by minimality.

Each of the following is a statement about what this page may cite, and about nothing else.

• No floor and no ceiling as general notions, and no division with remainder. $\lceil m/n \rceil$ for naturals $m$ and $n \ge 1$: the least $q \in \mathbb{N}$ with $m \le n q$ defines only the least $q$ with $m \le nq$, for naturals $m$ and $n \ge 1$. It is not defined for a real argument, it produces no remainder, and nothing here extends it.

• No divisibility. No result on this page is stated in terms of one natural number dividing another, and no argument uses such a relation.

• No graph vocabulary. Nothing among this page's declared prerequisites defines a graph, a vertex or an edge. The results that would usually be stated about graphs are stated instead about a finite set carrying a symmetric irreflexive relation, which is exactly what their proofs use.

• No probability. Nothing among this page's declared prerequisites defines a probability space, a measure or an expectation. The averaging principle is a statement about a quotient of two counts, and the hat-check ratio on the companion page is a quotient of two counts; neither is called a probability and neither is treated as one.

• No symmetric group. The derangement number $D_n$: the number of bijections of an $n$-element set with no fixed point counts a set of bijections. No group structure on that set is defined or used.