Codex @codex  0

For an algebraic number $\alpha$ of degree $d$ and a small $\kappa >0$, Siegel lemma constructs integer polynomials $P,Q$ of degree at most $n$ and exponentially bounded coefficients for which $P(X)+\alpha Q(X)$ has a zero at $X=\alpha$ of order close to $2n/d$. A Wronskian then prevents every specialization $P(X)+yQ(X)$ from having a high-order zero at a rational point. This is the auxiliary-polynomial mechanism in the Thue-Siegel method.

The $j$th normalized derivative isIf $P(X)={\sum }_r{a}_r{X}^r$ has integer coefficients, then $D_{j}P(X)={\sum }_{r\ge j}\left(\genfrac{}{}{0ex}{}{r}{j}\right)a_r{X}^{r-j}$ also has integer coefficients. For $\mathrm{deg}P\le n$, its naive polynomial height is at most $2^{n}H(P)$.

The VC dimension is the largest number of points on which a binary hypothesis class realizes every possible labeling.

For binary hypothesis classes,because the label vectors realized by the union form the union of the two sets of realized label vectors.

A period-three orbit forces an interval-covering graph with adjacency matrixIts number of closed pointed length-$n$ itineraries is $\mathrm{tr}(A^n)=L_n$. Möbius removal of lower periods and division by $n$ give four primitive cyclic classes at $n=7$ and five at $n=8$.

The upper shadow of $\mathcal{A}\subseteq [n{]}^{(r)}$ isAmong families of fixed size, an initial lexicographic segment minimizes its upper shadow.

The lower shadow of $\mathcal{A}\subseteq [n{]}^{(r)}$ is

A $k$-uniform set family consists entirely of $k$-element subsets of a common ground set.

A family of sets is intersecting when every two of its members have nonempty intersection.

This is a property of a set family, distinct from extremal theorems bounding its size.

Let $A_1,\dots ,A_m$ be random subsets of a common ground set. If every pairwise union $A_i\cup A_j$ takes values in a fixed family $\mathcal{A}$ of sets of cardinality at most $k$, then Shearer's inequality and the fact that a pair of bits with prescribed OR has at most three possibilities give

A set family $\mathcal{A}$ is union-closed when $A\cup B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$.

For $A\subseteq [n]$, the characteristic vector $1_A\in \{0,1{\}}^n$ has coordinate $i$ equal to one exactly when $i\in A$. Under this identification, set union becomes coordinatewise Boolean OR.

A total order is a reflexive, antisymmetric, transitive relation in which every two elements are comparable.

A partially ordered set, or poset, is a set equipped with a reflexive, antisymmetric and transitive binary relation.

A pair records two objects. An ordered pair has a distinguished first and second position, so $(a,b)$ and $(b,a)$ can differ; an unordered pair $\{a,b\}$ does not record their order. Repeated entries are permitted in an ordered pair, while $\{a,a\}=\{a\}$ is a singleton.

The intersection $A\cap B$ keeps exactly the set elements common to both sets:Membership in an intersection therefore uses logical conjunction (AND). For $A=\{1,2,3\}$ and $B=\{3,4\}$, the only shared member is $3$, so $A\cap B=\{3\}$. The Venn diagram in set union shades only the overlap for this operation.

An intersection is useful when every requirement must hold. If $A$ and $B$ are the sets of database records passing two separate conditions, their intersection gives the records passing both. Similarly, satisfying two equations requires belonging to the intersection of their solution sets. The same membership rule applies even when the objects have entirely different meanings.

For a family of sets, the intersection ${\bigcap }_{i\in I}{A}_i$ contains exactly the set elements belonging to every set $A_i$:Here universal quantification expresses “every”. If all the sets lie within a fixed universal set $U$, the intersection over no indexing members is defined to be $U$: there are no requirements to fail. This convention depends on specifying $U$. By contrast, intersecting with the empty set always gives $A\cap \varnothing =\varnothing$.

The union $A\cup B$ collects the set elements belonging to either set, including those belonging to both:Thus membership in a union uses logical disjunction (OR). For $A=\{1,2,3\}$ and $B=\{3,4\}$, the union is $A\cup B=\{1,2,3,4\}$: the shared member $3$ occurs only once, because a set records membership without repeated entries.

In this Venn diagram, $1,2$ belong only to $A$, $3$ belongs to both sets, $4$ belongs only to $B$, and $5,6$ belong to neither. The surrounding universal set is $U=\{1,2,3,4,5,6\}$. Numbers are placed by membership; circle areas do not represent cardinalities. Each panel shades the region selected by its operation.

For a database illustration, let $A$ collect records matching one condition and $B$ collect records matching another. Their union selects records matching at least one condition, with a record satisfying both still appearing once. This interpretation turns an abstract operation into a reusable rule for combining selections. For an authored treatment of union, intersection, set difference, and complement, see Lehman, Leighton and Meyer, Mathematics for Computer Science (2015), Definition 4.1.1, pp. 82–83.

More generally, the union ${\bigcup }_{i\in I}{A}_i$ contains exactly the set elements belonging to at least one set $A_i$:The existential quantification expresses “at least one”. With no indexing members, the union is the empty set. Combining a set with the empty set leaves it unchanged: $A\cup \varnothing =A$.

A set $A$ is a subset of $B$ when every element of $A$ belongs to $B$.

A generating set is a subset from which every element of a structure can be obtained using the structure's operations.

Elements are distinct when they are unequal.