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Two functions $f:A\to X$ and $g:B\to X$, whose domains lie in one ambient set, are equivalent modulo finite changes when $A△B$ is a finite set and the functions disagree at only finitely many points of $A\cap B$. The symmetric difference inclusion $A△C\subseteq (A△B)\cup (B△C)$ handles domains in a transitive chain. An exceptional set for values must also include points of $A\cap C$ absent from the middle domain $B$; these lie in the finite difference $A\setminus B$.

For a function $f:X\to Y$ and $y\in Y$, its fiber is the set $f^{-1}(y)=\{x\in X:f(x)=y\}$. Fibers partition the domain after empty fibers are omitted. A function is injective exactly when every fiber has at most one element. Quantum hidden subgroup problem algorithms use functions whose fibers are cosets.

For cross-intersecting families $\mathcal{A},\mathcal{B}$ with member sizes at most $r,s$ respectively, there is a finite set $X$ with $|X|\le r{\sum }_{j=0}^s{r}^j$ such that $A\cap B\cap X\ne \varnothing$ for every $A\in \mathcal{A},B\in \mathcal{B}$. Take a bounded-size transversal kernel ${\mathcal{A}}_0$ preserving hitting sets of size at most $s$ and set $X={\bigcup }_{A\in {\mathcal{A}}_0}A$. For each $B$, the set $B\cap X$ is a hitting set for ${\mathcal{A}}_0$, hence for all of $\mathcal{A}$. This proves the assertion even for infinite set families; if either family is empty, take $X=\varnothing$.

For $0<p<1$, every prescribed state pattern on a finite edge set has positive conditional probability given all exterior states: independence makes it $p^a(1-p{)}^b$ for $a$ prescribed open and $b$ prescribed closed edges. Thus a positive-probability exterior event remains possible after any finite local surgery. This is the finite-energy property of Bernoulli percolation, and applies equally to dual bond percolation.

A finite modification agrees with the identity map outside a finite exceptional set. Sorting that set gives a unique finite record of pairs $(x,f(x))$. Real-number pairing by separated digits encodes the record in $(0,1)$, and placing length-$n$ records in disjoint intervals $(n,n+1)$ gives an injection from all finite modifications into the real line. Conversely, changing the value at one fixed point gives a real-parameter family, so the Cantor-Schröder-Bernstein theorem shows this family has continuum cardinality.

A finite repetition-free sequence in a set $X$ is an injective function from an initial segment $\{0,\dots ,k-1\}$ of the natural numbers into $X$, including the empty sequence for $k=0$. For a finite set of size $m$, the number of these sequences isThe order of each list is part of the data, so it can be enumerated without choosing an order on its underlying support.

If $X$ is a Dedekind-finite set, its set of finite repetition-free sequences is also Dedekind-finite. An injective sequence of distinct finite lists would, by countable union of explicitly ordered finite lists without choice, either produce an injection $\mathbb{N}\to X$ or use only finitely many entries. The latter possibility is impossible because a fixed finite pool supports only finitely many repetition-free lists. If $X$ is infinite, the one-entry lists also show that the resulting set is infinite.

A finite set is a set with finitely many elements.

A function class is a set whose elements are functions, usually sharing a common domain and codomain.

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.

If a generic filter $G$ for an atomless forcing order belonged to its ground model, then the complement $\mathbb{P}\setminus G$ would be a dense ground-model set. A condition outside $G$ is already there; a condition in $G$ has two incompatible strengthenings, at least one outside its directed filter. Genericity would then require meeting the complement, a contradiction.

Hausdorff measure measures a set by the small-scale cost of covers, using the $s$th powers of their diameters and a standard normalization. In Euclidean dimension one, ${\mathcal{H}}^1$ agrees with arc length on smooth curves; in particular a line segment has its usual length.

A set $H$ is a hitting set for a set family $\mathcal{F}$ if $H\cap A\ne \varnothing$ for every $A\in \mathcal{F}$. Every set is a hitting set for an empty set family; none is a hitting set for a family containing the empty set. Bounded hitting sets permit finite certificates for some infinite set families, as in the bounded-size transversal kernel.

For a colouring $\chi :[S{]}^r\to \{0,\dots ,k-1\}$ of the $r$-element subsets of a set $S$, a subset $H\subseteq S$ is homogeneous if all its $r$-element subsets receive the same colour. The infinite Ramsey theorem for r-sets guarantees an infinite homogeneous subset when $S=\mathbb{N}$, $r$ is positive and the number of colours is finite. This does not guarantee a computable set when the colouring is computable.