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Additive combinatorics studies how the cardinalities and sumset representation functions of sumsets reveal structure in finite sets contained in abelian groups. Write the group operation as addition: the ambient abelian group may be the integers $\mathbb{Z}$, a finite cyclic group, or another abelian group. The same definitions apply, but the ambient group changes which bounds are possible.

The basic experiment is to add every ordered pair of members of a finite set. The sumset records the distinct answers; the sumset representation function records how often each answer occurs; the additive energy counts coincidences between those answers. An arithmetic progression produces many repeated sums, so its sumset grows slowly and its additive energy is relatively large. The doubling constant measures that growth relative to the original cardinality.

The examples below compare the arithmetic progression $A=\{0,1,2,3\}$ with the more spread-out integer set $B=\{0,1,4,10\}$. Their four members give sixteen ordered pairs in each case, but different numbers of distinct sums. This small comparison illustrates the definitions; it does not establish a claim about arbitrarily large sets. For a systematic introduction, see Tao and Vu's author page for Additive Combinatorics, with free sample chapters on sum set estimates and additive geometry.

The additive energy of a finite set $S\subseteq G$ in an abelian group $G$ counts additive quadruples, with order and repeated members allowed:At a fixed sum $s$, independently choosing two of its $r_{S+S}(s)$ ordered pairs gives $r_{S+S}(s{)}^2$ additive quadruples. Thus additive energy weights heavily represented sums more strongly than the cardinality of the sumset does.

For $A=\{0,1,2,3\}$ and $B=\{0,1,4,10\}$, the sumset representation functions giveThe arithmetic progression has additional coincidences such as $0+3=1+2$. The set $B$ still has coincidences forced by commutativity, including $0+1=1+0$, so its additive energy is not merely the sixteen identical-pair comparisons.

The same additive energy can be counted using the difference set. Define $r_{S-S}(t)=|\{(a,b)\in S\times S:a-b=t\}|$. The identity $a+b=c+d$ is equivalent to $a-c=d-b$, and the coordinate rearrangement $(a,b,c,d)\mapsto (a,c,d,b)$ is a bijection. ConsequentlyFor $A$, the difference counts from $-3$ through $3$ are again $1,2,3,4,3,2,1$. For $B$, the zero difference has four representations and each of the twelve nonzero differences has one, giving $4^2+12\cdot 1^2=28$. The sumset representation functions for sums and differences can look different while their squared totals agree.

For every nonempty finite set $S\subseteq G$ in an abelian group $G$,The lower bound follows from the Cauchy-Schwarz inequality and the total sumset representation function count:For the upper bound, any three members $a,b,c$ determine at most one fourth member $d=a+b-c$ of an additive quadruple. See Yufei Zhao's MIT lecture 25, pages 1-3, for representation counts and the energy bounds.

In the two integer examples, the lower bounds are $256/7$ and $256/10$, and the upper bound is $64$. For the subgroup $H=\{0,2,4\}$ of $\mathbb{Z}/6\mathbb{Z}$, each element of $H$ has three sum representations, so $E(H)=3\cdot 3^2=27$ and both bounds are equalities.

For nonempty $S$, dividing additive energy by $|S{|}^3$ gives a relative measure between $1/K(S)$ and one. Small doubling constant therefore forces relatively large additive energy. A converse requires care: the Balog-Szemerédi-Gowers theorem extracts a large subset with small doubling constant, rather than asserting that all of $S$ has small doubling constant.

An augmenting path in a matching $M$ is a nonempty alternating path in a matching with distinct endpoints that are both unmatched vertices in a matching. Its first and last edges lie outside $M$. Thus it has $k+1$ unselected edges and $k$ selected edges, for some $k\ge 0$. Taking the symmetric differenceremoves the selected edges on $P$ and inserts its unselected edges. Each internal vertex exchanges one selected edge for another; each endpoint gains its first selected edge. All choices away from $P$ stay in place, so $M^{\prime }$ is a matching in a graph with $|M^{\prime }|=|M|+1$.^[ref]

For an explicit repair, let the path graph $P_6$ have vertices $1,2,3,4,5,6$ in that order and start withThe endpoints $1$ and $6$ are initially unmatched vertices in a matching. Along $P$ the edges are outside, inside, outside, inside, outside $M$. Remove $(2,3)$ and $(4,5)$, then add $(1,2),(3,4),(5,6)$ together:This raises the cardinality from two to three and gives a perfect matching. Six vertices allow at most three disjoint pairs, so it is also a maximum cardinality matching. The initial $M$ is already a maximal matching: every unused edge touches a matched vertex. Augmentation succeeds because it can replace earlier choices instead of only adding to them.

One unused edge joining two unmatched vertices in a matching is the shortest possible augmenting path in a matching. Longer augmenting paths in a matching reassign several partners, explaining why a stalled greedy matching need not be optimal. For edge benefits $w_e$, the change in total benefit iswhich need not be positive even though the cardinality increases. A maximum-weight matching therefore needs a different optimality test. In an assignment equality graph an augmenting path in a matching moves toward a perfect matching while preserving zero reduced costs on matched edges; this is the setting used by the Hungarian algorithm.

A matroid basis is a matroid independent set that is a maximal element under inclusion of subsets: no additional element of the ground set can be admitted. Maximum cardinality means having the largest size among all matroid independent sets. Maximality tests whether this particular selection can grow; maximum size compares it with every selection. In a matroid the two conditions coincide, so all matroid bases have a common size. They need not have equal weight. ^[ref]

Proof outline. If a maximal matroid independent set $B$ were smaller than another matroid independent set $J$, the matroid independence axioms would admit an element of $J\setminus B$, contradicting maximality. Finiteness allows every matroid independent set to grow to a matroid basis by successive admissible additions. In a general independence system, maximal selections can have different sizes, as in the hereditary nonmatroid greedy obstruction.

For a finite simple graph $G$ and a matching in a graph $M\subseteq E(G)$, $M$ is a maximum cardinality matching if and only if $G$ has no augmenting path in a matching relative to $M$. The graph need not be a bipartite graph, and the assertion concerns cardinality, not edge benefits.^[ref]

The criterion turns a global comparison with every competing matching in a graph into a search for one repair. Starting from any matching in a graph, repeatedly find and toggle an augmenting path in a matching. Each step adds one selected edge, so finiteness gives termination after at most $\lfloor |V(G)|/2\rfloor -|M|$ steps; absence of any further augmenting path in a matching certifies maximum cardinality. This is an optimality criterion, not by itself a bound on the work needed to find each graph path.

For a short proof outline, an augmenting path in a matching immediately contradicts maximum cardinality. Conversely, if another matching in a graph $N$ has $|N|>|M|$, consider the subgraph whose set of edges is $M△N$. Its vertices have degree of a vertex at most two; its nontrivial connected components are graph paths and even cycles in a graph, alternating between the two matchings in a graph. A cycle in a graph contributes equal numbers from each. Hence some graph path contributes one more edge from $N$ than from $M$ and starts and ends with edges from $N$. Its endpoints are unmatched vertices in a matching relative to $M$, yielding the required augmenting path in a matching.

The cardinal numbers defined by ${\beth }_0={\aleph }_0$, ${\beth }_{\alpha +1}=2^{{\beth }_{\alpha }}$, and ${\beth }_{\lambda }={\sup }_{\alpha <\lambda }{\beth }_{\alpha }$ at limit ordinals are the beth numbers. In particular ${\beth }_1=|\mathcal{P}(\omega )|$, the cardinality of the continuum. This notation distinguishes iterated power sets from the enumeration by Aleph numbers of well-orderable infinite cardinal numbers.

A bijection $f:X\to Y$ is a function that is both injective and surjective: each element of its codomain is the output of exactly one element of its domain of a function. The two requirements exclude unused targets and merged inputs respectively. The finite example $f(A)=1$, $f(B)=2$, $f(C)=3$ is a bijection from $\{A,B,C\}$ to $\{1,2,3\}$, whereas adding an unused target $4$ makes that same rule fail surjectivity.

A bijection gives an inverse function by reversing every pairing. Conversely, a two-sided inverse function implies a bijection, as the proof under inverse function shows. This is useful for transporting structures without losing or identifying elements. It also defines equality of cardinality: two sets have equal cardinality exactly when a bijection pairs them. An infinite example is $n\mapsto 2n$ from the integers to the even integers, with inverse $m\mapsto m/2$ on that specified codomain. A proper subset can therefore have the same cardinality as an infinite set. ^[ref].

For a nonnegative integer $n$ and commuting quantities $x,y$,In expanding the $n$ factors, obtain $x^k{y}^{n-k}$ by selecting the $k$ positions that contribute $x$. There are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ such choices. For example,The coefficients are row five of Pascal's triangle, in reverse order under the displayed $x^k{y}^{n-k}$ convention; the row is symmetric. Setting $x=y=1$ gives $1+5+10+10+5+1=32$, the number of subsets of a five-element set. Each member has two inclusion choices, agreeing with the rule of product and the cardinality of the power set.

If there are injections $X\to Y$ and $Y\to X$, then there is a bijection $X\cong Y$.

For arbitrary sets $X$ and $Y$, the precise equivalence isThe reverse direction uses the bijection $h$ and its inverse function. Neither set is assumed to be a subset of the other: the hypothesis concerns functions, rather than containments. The forward direction holds in ZF without the axiom of choice.^[ref]

This makes cardinality comparison an antisymmetric relation: if each set is no larger than the other in the sense of an injection, they have equal cardinality. It also gives a practical strategy: build two convenient injections and let the theorem supply the bijection. The Injection-chain proof of the Cantor-Schröder-Bernstein theorem explains how to combine them without losing an output or pairing two inputs with the same output.

There is no surjection from a set $X$ onto its power set $\mathcal{P}(X)$. Consequently $\mathcal{P}(X)$ has strictly greater cardinality than $X$.

For any candidate function $f:X\to \mathcal{P}(X)$, form $D=\{x\in X:x\notin f(x)\}$. This is a subset of $X$ by set comprehension. If $f(d)=D$, then $d\in D$ holds exactly when $d\notin D$, a contradiction. Thus $D$ is missing from the image of $f$. On the other hand, $x\mapsto \{x\}$ is an injection from $X$ into its power set, establishing the strict comparison of cardinalities. For $X=\mathbb{N}$, the omitted selection is the subset version of Cantor's diagonal argument.

A capture-recapture model is a statistical model for repeated observations of identifiable individuals, including those missed on some occasions. A closed-population capture-recapture model estimates a fixed population cardinality; models allowing individuals to enter or leave can instead describe survival and recruitment. The sampling assumptions determine which likelihood function is appropriate.

For the two-sample closed-population capture-recapture model, condition on the sample sizes $M$ and $C$. If the second sample selects $C$ distinct individuals from $N$ by simple random sampling, then the overlap $R$ has the hypergeometric distribution:Here $r$ is an integer and the probability mass function is zero outside these bounds. The binomial coefficients count choices of $r$ marked and $C-r$ unmarked individuals, divided by all equally likely size-$C$ subsets. The hypergeometric distribution has expected valueEquating the observed overlap with this expected value gives the Lincoln-Petersen estimator. The underlying sampling without replacement, probability mass function, and expected value are developed in Raymond P. Guiteras, MIT 14.30 hypergeometric distribution handout, §§1–4 and 8.

For the toy counts $M=20$, $C=25$, and $R=5$, hold the observations fixed and vary the unknown integer $N$. The likelihood function isValues below forty cannot contain the observed set union. Exact comparison of consecutive likelihoods gives two tied maximum-likelihood estimates, $N=99$ and $N=100$. The Lincoln-Petersen estimator returns $100$; the Chapman estimator returns $90$ and is a different rule for estimation.

The vertical scale is $L(N)/{\max }_{n\ge 40}L(n)$. This relative likelihood compares how well candidate statistical parameters explain the same observations; it is not a posterior probability. The displayed range $40\le N\le 300$ is only a plotting window. The likelihood function remains positive for every integer $N\ge 40$ and tends to zero as $N$ increases, so the curve has a continuing right tail. The data do not justify truncating statistical inference at three hundred. The conditional hypergeometric distribution formulation also appears in Louis-Paul Rivest and Mamadou Yauck, Small Sample Estimators for Two-way Capture Recapture Experiments, §3, equation (5).

Zero recaptures require care. With $R=0$ and positive $M,C$, the Lincoln-Petersen estimator has no finite ratio, but the capture-recapture model still gives a valid likelihood function:This likelihood function increases toward one as $N$ grows, so there is no finite maximum-likelihood estimate. For the same sample sizes twenty and twenty-five with zero overlap, forty-five distinct individuals would be observed. Lack of overlap supplies a lower bound but very weak information about large population cardinalities.

In more elaborate capture-recapture models, a first-recapture array $m_{ij}$ counts individuals released at occasion $i$ and next observed at $j$. A probability of survival ${\varphi }_j$ and a probability of detection $P_{j+1}$ contribute a no-observation continuation factor ${\varphi }_j(1-P_{j+1})$. The resulting likelihood function depends on both survival and detection; these probabilities need not be separately identifiable at the last observation interval. Missing-count EM for capture-recapture describes one way to fit such a model, and identifiability remains a separate issue from computing an estimator output.

An increasing chain of countable sets has union of cardinality at most ${\aleph }_1$. If the union is uncountable, select ${\aleph }_1$ of its points and one containing member for each. Every countable member misses one selected point, so linear comparison puts it below that point's chosen member. The chosen subfamily is cofinal, and its union has size at most ${\aleph }_1$ by infinite cardinal arithmetic.

The cardinality $|X|$ of a set $X$ expresses its size up to bijection. Two sets have the same cardinality exactly when a bijection pairs their members. For a finite set, cardinality is the ordinary number of set elements. In the presence of the axiom of choice, every set can be assigned the least ordinal in its bijection class as its cardinal number.

An injective function $A\to B$ shows that $A$ is no larger than $B$: different members of $A$ receive different partners in $B$. If there are injections in both directions, the Cantor-Schröder-Bernstein theorem supplies a bijection. Infinite cardinality records this pairing structure rather than a geometric dimension or density. That is why a line of natural numbers and a grid of natural-number pairs can have the same size, while an interval of real numbers has larger cardinality.

Write $r_3(N)$ for the maximum cardinality of a subset of $[N]$ containing no nonconstant three-term arithmetic progression. There is an absolute constant $C$ such that every three-term-progression-free subset $A\subseteq [N]$ satisfies $|A|\le CN/\log \log N$ for $N\ge 3$. A quantitative Roth density-increment step gives, for subset density $\delta$ and $N\ge 4\cdot 10^6{\delta }^{-4}$, a arithmetic progression of length at least ${\delta }^2\sqrt{N}/2000$ and increased subset density at least $\delta +{\delta }^2/64$. The increment decreases reciprocal subset density by at least $1/65$, so fewer than $\lceil 65/\delta \rceil$ iterations are possible. If $N_j$ is the successive length, then $\log N_{j+1}\ge \frac{1}{2}\log N_j-\log (2000/{\delta }^2)$, using the initial subset density to bound all later ones. Hence $\log N_j\ge 2^{-j}\log N-2\log (2000/{\delta }^2)$. If $\delta \log \log N$ exceeds a sufficiently large absolute constant, all these intervals remain above the density increment threshold, a contradiction. This proves the displayed bound. The result is a classical quantitative version of Roth theorem on three-term arithmetic progressions, without asserting an optimal bound.

A closed-population capture-recapture model assumes the same individuals remain throughout the observations: no births, deaths, immigration, or emigration. Its elementary two-sample version additionally requires retained, correctly recognized marks and a second sample chosen by simple random sampling independently of first-sample membership. Mixing and unchanged capture behavior help make this plausible. Equal individual capture probabilities alone do not guarantee that every size-$C$ subset is equally likely; that stronger sampling assumption gives the hypergeometric distribution. See Barker, §§1.1.1 and 1.1.3: closure, marks, and catchability assumptions.

Unequal catchability, mark loss, and population change can invalidate this capture-recapture model. For example, if especially catchable individuals appear repeatedly, overlap can be inflated and the Lincoln-Petersen estimator can underestimate population cardinality. Lost marks can decrease recorded overlap and increase the same ratio estimator, all else equal. Two samples alone cannot establish that the sampling assumptions hold.

Complementing all members preserves cardinality and gives $|A^c\cap B^c|=n-2r+|A\cap B|$. Thus complementation bijects t-intersecting families of $r$-sets with $(n-2r+t)$-intersecting families of $(n-r)$-sets. It swaps left and right coordinate shifts of a set family. When the transformed intersection parameter is positive, extremal families and their generating-support lemmas can therefore be compared in opposite compression directions.

A connected component of a topological space $X$ is a nonempty connected subset that is maximal under subset inclusion. If $C$ is a connected component and $C\subseteq D\subseteq X$ with $D$ a connected subset, then $D=C$. Maximality means that no larger connected subset contains $C$; it does not mean largest cardinality among all connected subsets.

For example, the connected components of $[-2,2]\setminus \{0\}$ in the usual subspace topology of the real line are $[-2,0)$ and $(0,2]$. Each is a real interval, and no connected subset of the real line can meet both without containing the missing point $0$. For the finite set $\{-2,0,2\}$ in the same subspace topology, the three connected components are its singleton sets.

Every point $x\in X$ lies in exactly one connected component, constructed by taking the unionAll members contain $x$, so their union is connected by connected union with a connected hub. The union includes every connected subset through $x$, so it is maximal. Equivalently, defineThis is an equivalence relation: singleton sets give reflexivity, symmetry is immediate, and two connected subsets sharing the middle point have a connected union, giving transitivity. The equivalence classes are precisely the connected components. They form a partition of a set: they cover $X$ and any two are either equal or disjoint.^[ref]

Every nonempty connected subset lies within one connected component. Moreover, every connected component is closed in $X$: closure of a connected set makes its closure connected, and maximality forces that closure to equal the original connected component. A connected component need not be open. For example, in $S=\{0\}\cup \{1/n:n=1,2,\dots \}$ with its usual subspace topology from the real line, all connected components are singleton sets, but $\{0\}$ is not open: every neighbourhood of $0$ in $S$ contains some $1/n$.^[ref]

For distinct coordinates $i,j$, replace a member containing $j$ but not $i$ by its partner with $j$ exchanged for $i$, provided that partner is absent. This set family operation preserves cardinality. Repeating shifts with $i<j$ terminates because every change decreases ${\sum }_{A\in \mathcal{F}}{\sum }_{a\in A}a$. Its effect on shadows is described by coordinate-shift shadow containment.

Assume exactly two operating-system processes, $N$ initially resident private memory pages of $P$ bytes each, initially distinct page frames shared by A and B. Neither unmaps, exits, forks again, or encounters page replacement; no other owner exists, allocation succeeds, and a first write copies one whole shared memory page. Once a page frame is exclusively owned it is reused on writes, without another copy; writable mappings remain writable. Count data storage only, excluding page tables and other bookkeeping. With B as the only writer, writing $k$ distinct memory pages givesThis depends on distinct written memory pages, not the number of store instructions or reads. For the diagrams, take $N=3$ and an illustrative $P=4096$ bytes:

For both writers, let $S_A,S_B$ be the sets of initial memory page numbers written by A and B. Under these assumptions, new allocations and copies both equal $|S_A\cup S_B|$, the cardinality of their set union, while resident data storage is $(N+|S_A\cup S_B|)P$. Each initially shared page frame needs at most one split: after the first writer copies it, the other owns the original exclusively. Both writing the same memory page therefore produces two versions with one copy, even if the final values happen to match. The last row can incur two protection page faults, one for each writer, while making only one copy. A handler that always allocates on a protected write can make a second copy and free the now-unreferenced original: allocation counts and final resident storage are different observables.

These are model counts, not measured Linux costs or a guaranteed bound on total computer memory. More descendants, other owners, demand paging, page replacement, different memory page sizes, and different exclusive-ownership rules change the accounting. The useful claim is conditional: deferred data copying can cost in proportion to the distinct shared memory pages that are written, while creating separate page tables has its own cost.

// Target: computer-science.bigb

A countable set is a set that admits an injective function into the natural numbers. It is therefore either a finite set or a countably infinite set.

For a countably infinite set, countability means that its members can be arranged as a sequence in which every member occurs exactly once. The order need not resemble the order already present in the set. For example, the integers can be listed as $0,1,-1,2,-2,\dots$. Simply increasing through positive values would never reach a negative one.

The Cartesian product ${\mathbb{N}}^2$ needs a similar change of strategy. Listing a whole horizontal row first would never finish that row, so later rows would be missed. Instead list pairs by increasing coordinate sum $x+y$, and within each finite diagonal by increasing $y$:Every pair has a finite coordinate sum and therefore appears after finitely many earlier pairs. This is the construction made explicit by the Cantor pairing function. It explains why having two infinite coordinates does not, by itself, mean a larger cardinality.