A Venn diagram represents sets by closed curves, with each region recording a possible pattern of membership. For two sets, the regions mean “first only”, “both”, “second only”, and “neither”; the enclosing rectangle can represent the universal set. Shading regions displays union, intersection, set difference, or complement. In the numbered examples in set union and complement, positions encode membership, while curve shapes and region areas do not encode cardinalities.

A subset $A$ of a specified ambient set $X$ is coinfinite if its complement of a set $X\setminus A$ is infinite. In $\mathbb{N}$, the even numbers are both infinite and coinfinite, whereas a cofinite set is not coinfinite.

A partition of a set $X$ is a family of pairwise disjoint nonempty subsets whose union is $X$. Every equivalence relation defines a partition into equivalence classes, and conversely belonging to the same cell of a partition defines an equivalence relation. The nonempty colour classes of a finite colouring form a finite partition.

A set element is an object belonging to a set. The statement $a\in A$ asserts membership; $a\notin A$ denies it. Membership differs from inclusion: $a\in A$ concerns one member, whereas $B\subseteq A$ asserts that every set element of $B$ belongs to $A$.

A cover of a set $E$ is a family of sets whose union contains $E$. The members may overlap and may include points outside $E$; intersecting each member with $E$ gives a cover by its subsets. An open cover requires its members to be open sets. In Hausdorff measure, countable covers with a bound on each member's diameter determine geometric size. Constructing one inexpensive cover supplies an upper bound, while a lower bound must control all allowed covers.

The solution set of a mathematical condition $P$ in a specified domain $D$ collects every allowed value that satisfies it:It describes the answer without committing to the steps used to find it. For the equation $x^2=1$ over the real numbers, the solution set is $\{-1,1\}$; restricting to positive real numbers leaves $\{1\}$. For $2x=1$, the solution set is $\{1/2\}$ over the rational numbers and the empty set over the integers. The domain is part of the question.

An equation can have one solution, several solutions, no solutions, or an entire domain of solutions. Over the real numbers, $2x+3=9$ gives the singleton set $\{3\}$, $x^2+1=0$ gives the empty set, and the mathematical identity $(x+1{)}^2=x^2+2x+1$ gives the whole real line. Changing that second equation's domain to the complex numbers gives $\{-i,i\}$. The elementary definition by values that satisfy an equation and the classification by solution sets appear in College Algebra 2e, §2.2: Solving Linear Equations in One Variable.

Requiring several conditions simultaneously takes the intersection of their solution sets. The squared equation $x=(x-2{)}^2$ has solution set $\{1,4\}$. Adding the restriction $x\ge 2$ giveswhich is also the solution set of $\sqrt{x}=x-2$. The extraneous solution example explains why the sign restriction is needed. Requiring at least one of several conditions instead takes their union. This turns logical combinations of constraints into reusable operations on sets.

For several variables, a solution is an ordered assignment, usually represented by a tuple. The real equation $x+y=1$ has solution set $\{(t,1-t):t\in \mathbb{R}\}$, a straight line in the Euclidean plane with coordinates $(x,y)$. Adding $x-y=0$ intersects that straight line with another, leaving the single tuple $(1/2,1/2)$. A system of linear inequalities has no solution exactly when the intersection of its individual solution sets is the empty set. For optimization constraints, this common solution set is the feasible region.

Because a solution set is a set, the order in which its members are listed is irrelevant. The multiple root of $(x-1{)}^2=0$ is still the one solution $1$; its multiplicity of a root is additional polynomial information, rather than a second occurrence inside the solution set. Equivalent equations preserve the solution set on the stated domain, even if their algebraic forms differ.

For a Hermitian form or a Hermitian form over a quadratic finite field, the displayed subspace is the orthogonal complement of $W$. If the form restricts nondegenerately to a finite-dimensional $W$, its invertible Gram matrix supplies, for each $v$, a unique $w_0\in W$ with $B(v-w_0,W)=0$. Consequently $V=W\oplus W^{\perp }$. When the form on $V$ is nondegenerate, its restriction to $W^{\perp }$ is too: a radical vector there is orthogonal to both summands and hence is zero.

For a real subadditive sequence $(x_n{)}_{n\ge 1}$, the normalized sequence has the extended-real limit of a sequenceNo lower-bound hypothesis is needed for this conclusion. The limit of a sequence is finite precisely when the ratios are bounded below. The example $x_n=-n^2$ is a subadditive sequence with normalized limit of a sequence $-\infty$.

To prove the result, fix $k\ge 1$ and, for $n\ge k$, write $n=qk+r$ with $q\ge 1$ and $0\le r<k$. Iterating subadditivity gives $x_n\le q{x}_k+C_k$, where $C_k=\max (0,x_1,\dots ,x_{k-1})$ and $C_1=0$. Hence ${\mathrm{limsup}}_n{x}_n/n\le x_k/k$ for every $k$. Every ratio is at least their infimum. If that infimum is finite these bounds identify the limit of a sequence; if it is $-\infty$, choosing $k$ with an arbitrarily negative ratio gives convergence to $-\infty$.

A series adds the terms of a sequence in their specified order. Its infinite-sum notation is defined through finite partial sums:when the displayed limit of a sequence exists as a finite number. Otherwise the series is not a convergent series in the ordinary sense. Writing an infinite list of additions does not by itself assign it a finite sum. Cesàro summation is a different prescription, with its own definition.

Two positive series make the distinction concrete. For $a_n=1/n$, the individual terms tend to zero but the harmonic series has unbounded partial sums. For $a_n=1/[n(n+1)]$, partial fraction decomposition reveals a telescoping series whose partial sums tend to one. The useful question is how much all the remaining terms can contribute together, rather than only how small the next term looks.

The Lucas numbers obey the same second-order recurrence as the Fibonacci numbers but start with $2,1$, so their sequence is $2,1,3,4,7,11,\dots$. For $n\ge 1$, $L_n=F_{n-1}+F_{n+1}$. Substitution in the Fibonacci addition formula gives $F_{2n}=F_n{L}_n$. Consecutive Fibonacci numbers are coprime by the Euclidean algorithm; since $L_n=F_n+2F_{n-1}$, Bézout's identity gives $\gcd (F_n,L_n)=\gcd (F_n,2)\le 2$.

For $0\le r<2^n$ and $n\ge 0$, the displayed sequence tends to zero, but its series diverges. The signed sum across the $n$th dyadic block has magnitude ${\sum }_{r=0}^{2^n-1}(2^n+r{)}^{-1}\ge 1/2$. Differences of arbitrarily late partial sums therefore fail the Cauchy sequence condition on partial sums. Alternating signs between blocks do not rescue convergence when the total contribution of an entire block stays away from zero.

After testing nonnegative edge coordinates and any explicit box or budget inequalities, treat those coordinates as lengths. A shortest path of length below one supplies a violated path inequality; a minimum length at least one certifies all path inequalities. If no source-to-target path exists, all those inequalities are vacuous. Thus a polynomial time shortest path problem calculation separates an exponentially constrained linear program.

A heat equation with a nonlinear reaction term depending on the value of its solution.

If $N$ is an elementary abelian normal subgroup and $|G/N|$ is prime to its characteristic, the extension splits and its complements are conjugate by $N$. In additive notation an extension cocycle satisfies $f(h,k)+f(hk,t)=hf(k,t)+f(h,kt)$. Averaging over $t$ gives $f(h,k)=b(h)+hb(k)-b(hk)$, where $b(h)=|G/N{|}^{-1}{\sum }_{t}f(h,t)$. Changing the section by $b$ removes the cocycle. For two complements the difference cocycle has $d(hk)=d(h)+hd(k)$; averaging gives $d(h)=b-hb$, which is conjugation by an element of $N$. These arguments use invertibility of the group order on $N$ and underlie Hall conjugacy in soluble groups.

If $N=⟨A\mid R⟩$ and $H=⟨B\mid S⟩$ have finite group presentations and $\varphi :H\to \mathrm{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\varphi (b)(a)$. Then $N{⋊}_{\varphi }H$ has the finite presentationThe action automorphisms also let these relations rewrite conjugation by inverse generators. Every word becomes an $A$-word followed by a $B$-word. The natural map to the semidirect product is injective because its triviality forces the $B$-word to be trivial using $S$ and then the $A$-word to be trivial using $R$.

Let an abelian group $V$ carry an action $\rho$ of a group $H$ by automorphisms, and use multiplication $(h,v)(k,w)=(hk,v+\rho (h)w)$. Comparing $(e,v)(h,w)$ and $(h,w)(e,v)$ shows that a translation is central precisely when its vector is fixed by the entire action. If $H$ is itself abelian, the whole semidirect product is abelian exactly when the action is trivial. For $H=(\mathbb{R},+)$, $V={\mathbb{R}}^2$, and determinant-one linear actions fixing the first coordinate vector, $\rho (x)=\left(\begin{array}{cc}1& r(x)\\ 0& 1\end{array}\right)$, where $r$ is an additive function. Without a regularity assumption, it need not be $r(x)=cx$.

For a finitely generated module $V$ on the right over a semilocal ring, the displayed condition makes the induced module endomorphism on $V/VJ$ injective. This quotient is an Artinian module, so Artinian modules are co-Hopfian makes that map surjective. The remaining quotient module $V/\alpha (V)$ equals its product with $J$, and Nakayama lemma makes it zero.

A finite family of contracting similarities $S_1,\dots ,S_k$ satisfies the open set condition if there is a nonempty bounded open set $O$ with $S_i(O)\subseteq O$ and $S_i(O)\cap S_j(O)=\varnothing$ for $i\ne j$. This permits the closures to meet. For the invariant compact set, this separation makes its Hausdorff dimension equal to the similarity exponent $s$ defined by ${\sum }_i{c}_i^s=1$, where $c_i$ are the contraction ratios. The level-$n$ cylinder sets supply the upper dimension bound; the separated cylinder masses with weights $c_i^s$ supply the matching mass-distribution lower bound. For the 24 maps of ratio $1/5$ retaining all but the central square, $O=(0,1{)}^2$ works and $s=\log 24/\log 5$.

A contracting Euclidean similarity multiplies every Euclidean distance by a constant $c<1$; equivalently it has the form $cRx+b$ for an orthogonal matrix $R$ and translation vector $b$. Finite families of such maps generate compact sets with self-similarity.

For finitely many contracting Euclidean similarities, the map $\mathcal{T}$ on nonempty compact subsets of ${\mathbb{R}}^n$ is a contraction in the Hausdorff metric, with contraction constant ${\max }_i{c}_i<1$. The contraction mapping theorem gives a unique nonempty compact invariant set. If the open set condition holds, its Hausdorff dimension is the unique $s$ satisfying ${\sum }_i{c}_i^s=1$.