hcju/proofwikips huggingface dataset:
Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal/Reverse Implication
Let $\left({D, +, \circ}\right)$ be a principal ideal domain. Let $\left({p}\right)$ be the principal ideal of $D$ generated by $p$. Let $\left({p}\right)$ be a maximal ideal of $D$. Then $p$ is irreducible.
Subring of Polynomials over Integral Domain Contains that Domain
Let $\struct {R, +, \circ}$ be a commutative ring. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Let $D \sqbrk x$ denote the ring of polynomials in $x$ over $D$. Then $D \sqbrk x$ contains $D$ as a subring and $x$ as an element.
Subring of Polynomials over Integral Domain is Smallest Subring containing Element and Domain
Let $\struct {R, +, \circ}$ be a commutative ring. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Let $D \sqbrk x$ denote the ring of polynomials in $x$ over $D$. Then $D \sqbrk x$ is the smallest subring of $R$ which contains $D$ as a subring and $x$ as an element.
Cantor-Bernstein-Schröder Theorem/Lemma
Let $S$ be a set. Let $T \subseteq S$. Suppose that $f: S \to T$ is an injection. Then there is a bijection $g: S \to T$.
Cantor-Bernstein-Schröder Theorem/Proof 5
Let $S$ and $T$ be sets. Let $f: S \to T$ and $g: T \to S$ be injections. Then there exists a bijection $\phi: S \to T$.
Quotient Space of Real Line may be Indiscrete
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\Q$ be the rational numbers. Let $\mathbb I$ be the irrational numbers. Then $\set {\Q, \mathbb I}$ is a partition of $\R$. Let $\sim$ be the equivalence relation induced on $\R$ by $\set {\Q, \mathbb I}$. Let $T_\sim := \struct {\R / {\sim}, \tau_\sim} $ be the quotient space of $\R$ by $\sim$. Then $T_\sim$ is an indiscrete space.
Quotient Space of Real Line may not be Kolmogorov
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then there exists an equivalence relation $\sim$ on $\R$ such that the quotient space $\struct {\R / {\sim}, \tau_\sim}$ is not Kolmogorov.
Quotient Space of Real Line may be Kolmogorov but not Fréchet
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Define an equivalence relation $\sim$ by letting $x \sim y$  either: :$x = y$ or: :$x, y \in \Q$ Let $\struct {\R / {\sim}, \tau_\sim}$ be the quotient space of $\R$ by $\sim$. Then $\struct {\R / {\sim}, \tau_\sim}$ is a Kolmogorov space but not a Fréchet space.
Unique Representation in Polynomial Forms/General Result
Let $f$ be a polynomial form in the indeterminates $\set {X_j: j \in J}$ such that $f: \mathbf X^k \mapsto a_k$. For $r \in \R$, $\mathbf X^k \in M$, let $r \mathbf X^k$ denote the polynomial form that takes the value $r$ on $\mathbf X^k$ and zero on all other monomials. Let $Z$ denote the set of all multiindices indexed by $J$. Then the sum representation: :$\ds \hat f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ has only finitely many non-zero terms. Moreover it is everywhere equal to $f$, and is the unique such sum.
Law of Excluded Middle for Two Variables
:$\vdash (p \land q) \lor (\lnot p \land q) \lor (p \land \lnot q) \lor (\lnot p \land \lnot q)$
Ring of Polynomial Forms is Commutative Ring with Unity
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $A = R \sqbrk {\set {X_j: j \in J} }$ be the set of all polynomial forms over $R$ in the indeterminates $\set {X_j: j \in J}$. Then $\struct {A, +, \circ}$ is a commutative ring with unity.
Idempotent Elements form Subsemigroup of Commutative Semigroup
Let $\struct {S, \circ}$ be a semigroup such that $\circ$ is commutative. Let $I$ be the set of all elements of $S$ that are idempotent under $\circ$. That is: :$I = \set {x \in S: x \circ x = x}$ Then $\struct {I, \circ}$ is a subsemigroup of $\struct {S, \circ}$.
Product of Commuting Idempotent Elements is Idempotent
Let $\struct {S, \circ}$ be a semigroup. Let $a, b \in S$ be idempotent elements of $S$. Let $a$ and $b$ commute: :$a \circ b = b \circ a$ Then $a \circ b$ is idempotent.
Set of All Self-Maps is Semigroup
Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself. Let the operation $\circ$ represent composition of mappings. Then the algebraic structure $\struct {S^S, \circ}$ is a semigroup.
Integers form Subdomain of Reals
The integral domain of integers $\struct {\Z, +, \times}$ forms a subdomain of the field of real numbers.
Identity Element is Idempotent
Let $\struct {S, \circ}$ be an algebraic structure. Let $e \in S$ be an identity with respect to $\circ$. Then $e$ is idempotent under $\circ$.
Identity of Algebraic Structure is Preserved in Substructure
Let $\struct {S, \circ}$ be an algebraic structure with identity $e$. Let $\struct {T, \circ}$ be a algebraic substructure of $\struct {S, \circ}$. That is, let $T \subseteq S$. Let $e \in T$. Then $e$ is an identity of $\struct {T, \circ}$.
Idempotent Elements form Submonoid of Commutative Monoid
Let $\left({S, \circ}\right)$ be a commutative monoid. Let $e \in S$ be the identity element of $\left({S, \circ}\right)$. Let $I$ be the set of all elements of $S$ that are idempotent under $\circ$. That is: :$I = \left\{{x \in S: x \circ x = x}\right\}$ Then $\left({I, \circ}\right)$ is a submonoid of $\left({S, \circ}\right)$ with identity $e$.
Inverse Image under Order Embedding of Strict Upper Closure of Image of Point
Let $\struct {S, \preceq}$ and $\struct {T, \preceq'}$ be ordered sets. Let $\phi: S \to T$ be an order embedding of $\struct {S, \preceq}$ into $\struct {T, \preceq'}$ Let $p \in S$. Then: :$\map {\phi^{-1} } {\map \phi p^{\succ'} } = p^\succ$ where $\cdot^\succ$ and $\cdot^{\succ'}$ represent strict upper closure with respect to $\preceq$ and $\preceq'$, respectively.
Equivalence of Definitions of Order Embedding/Definition 1 implies Definition 3
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. Let $\phi: S \to T$ be an order embedding by Definition 1: {{:Definition:Order Embedding/Definition 1}} Then $\phi: S \to T$ is an order embedding by Definition 3: {{:Definition:Order Embedding/Definition 3}}
Equivalence of Definitions of Order Embedding/Definition 3 implies Definition 1
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. Let $\phi: S \to T$ be an order embedding by Definition 3: {{:Definition:Order Embedding/Definition 3}} Then $\phi: S \to T$ is an order embedding by Definition 1: {{:Definition:Order Embedding/Definition 1}}
Path as Parameterization of Contour
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path. Let there exist $n \in \N$ and a subdivision $\left\{{a_0, a_1, \ldots, a_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$ such that: : $\gamma {\restriction_{ \left[{a_{k - 1} \,.\,.\, a_k}\right] } }$ is a smooth path for all $k \in \left\{ {1, \ldots, n}\right\}$ where $\gamma {\restriction_{\left[{a_{k - 1} \,.\,.\, a_k}\right]} }$ denotes the restriction of $\gamma$ to $\left[{a_{k - 1} \,.\,.\, a_k}\right]$. Then there exists a contour $C$ with parameterization $\gamma$ and these properties: :$(1): \quad$ If $\gamma$ is a closed path, then $C$ is a closed contour. :$(2): \quad$ If $\gamma$ is a Jordan arc, then $C$ is a simple contour. :$(3): \quad$ If $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.
Kernel of Induced Homomorphism of Polynomial Forms
Let $R$ and $S$ be commutative rings with unity. Let $\phi: R \to S$ be a ring homomorphism. Let $K = \ker \phi$. Let $R \left[{X}\right]$ and $S \left[{X}\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$. Let $\bar\phi: R \left[{X}\right] \to S \left[{X}\right]$ be the induced morphism of polynomial rings. Then the kernel of $\bar\phi$ is: :$\ker \bar\phi = \left\{{ a_0 + a_1 X + \cdots + a_n X^n \in R \left[{X}\right] : \phi \left({a_i}\right) = 0 \text{ for } i = 0, \ldots, n }\right\}$ Or, more concisely: :$\ker \bar\phi = \left({\ker \phi}\right) \left[{X}\right]$
Boundary of Polygon as Contour
Let $P$ be a polygon embedded in the complex plane $\C$. Denote the boundary of $P$ as $\partial P$. Then there exists a simple closed contour $C$ such that: : $\operatorname{Im} \left({C}\right) = \partial P$ where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.
Zero Simple Staircase Integral Condition for Primitive
Let $f: D \to \C$ be a continuous complex function, where $D$ is a connected domain. Let $\displaystyle \oint_C \map f z \rd z = 0$ for all simple closed staircase contours $C$ in $D$. Then $f$ has a primitive $F: D \to \C$.
Preordering induces Equivalence Relation
Let $\struct {S, \precsim}$ be a preordered set. Define a relation $\sim$ on $S$ by letting $x \sim y$  $x \precsim y$ and $y \precsim x$. Then $\sim$ is an equivalence relation.
Content of Rational Polynomial is Multiplicative
Let $h \in \Q \sqbrk X$ be a polynomial with rational coefficients. Let $\cont h$ denote the content of $h$. Then for any polynomials $f, g \in \Q \sqbrk X$ with rational coefficients: :$\cont {f g} = \cont f \cont g$
Gauss's Lemma on Irreducible Polynomials
Let $\Z$ be the ring of integers. Let $\Z \sqbrk X$ be the ring of polynomials over $\Z$. Let $h \in \Z \sqbrk X$ be a polynomial.  :$(1): \quad h$ is irreducible in $\Q \sqbrk X$ and primitive :$(2): \quad h$ is irreducible in $\Z \sqbrk X$.
Antisymmetric Quotient of Preordered Set is Ordered Set
Let $\struct {S, \precsim}$ be a preordered set. Let $\sim$ be the equivalence relation on $S$ induced by $\precsim$. Let $\struct {S / {\sim}, \preceq}$ be the antisymmetric quotient of $\struct {S, \precsim}$. Then: :$\struct {S / {\sim}, \preceq}$ is an ordered set. :$\forall P, Q \in S / {\sim}: \paren {P \preceq Q} \land \paren {p \in P} \land \paren {q \in Q} \implies p \precsim q$ This second statement means that we could just as well have defined $\preceq$ by letting $P \preceq Q$ iff: :$\forall p \in P: \forall q \in Q: p \precsim q$
Ordering on Partition Determines Preordering
Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\phi: S \to \PP$ be the quotient mapping. Let $\preceq$ be a ordering of $\PP$. Define a relation $\precsim$ on $S$ by letting $p \precsim q$ : :$\map \phi p \preceq \map \phi q$ Then: :$\precsim$ is a preordering on $S$. :$\precsim$ is the only preordering on $S$ that induces the $\preceq$ ordering on $\PP$.
Units of Ring of Polynomial Forms over Commutative Ring
Let $\struct {R, +, \circ}$ be a non-null commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $R \sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $R$. Let $\map P X = a_0 + a_1 X + \cdots + a_n X^n \in R \sqbrk X$. Then: :$\map P X$ is a unit of $R \sqbrk X$ : :$a_0$ is a unit of $R$ Also, for $i = 1, \ldots, n$, $a_i$ is nilpotent in $R$.
Polynomials Closed under Addition/Polynomials over Integral Domain
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity. Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$. Then $\forall x \in R$, the set $D \left[{x}\right]$ of polynomials in $x$ over $D$ is closed under the operation $+$.
Polynomials Closed under Addition/Polynomials over Ring
Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +, \circ}\right)$ be a subring of $R$. Then $\forall x \in R$, the set $S \left[{x}\right]$ of polynomials in $x$ over $S$ is closed under the operation $+$.
Nilpotent Element is Zero Divisor
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Suppose further that $R$ is not the null ring. Let $x \in R$ be a nilpotent element of $R$. Then $x$ is a zero divisor in $R$.
Units of Ring of Polynomial Forms over Integral Domain
Let $\struct {D, +, \circ}$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in an indeterminate $X$ over $D$. Then the group of units of $D \sqbrk X$ is precisely the group of elements of $D \sqbrk X$ of degree zero that are units of $D$.
Kernel of Magma Homomorphism is Submagma
Let $\left({S, *}\right)$ and $\left({T, \circ}\right)$ be algebraic structures. Let $\left({T, \circ}\right)$ have an identity $e$. Let $\phi: S \to T$ be a magma homomorphism. Then the kernel of $\phi$ is a submagma of $\left({S, *}\right)$. That is: :$\left({\phi^{-1} \left({e}\right), *}\right)$ is a submagma of $\left({S, *}\right)$ where $\phi^{-1} \left({e}\right)$ denote the preimage of $e$.
Preimage of Zero of Homomorphism is Submagma
Let $\struct {S, *}$ be a magma. Let $\struct {T, \circ}$ be a magma with a zero element $0$. Let $\phi: S \to T$ be a magma homomorphism. Then $\struct {\phi^{-1} \sqbrk 0, *}$ is a submagma of $\struct {S, *}$.
Polynomial over Field is Reducible iff Scalar Multiple is Reducible
Let $K$ be a field. Let $K \left[{X}\right]$ be the ring of polynomial forms over $K$. Let $P \in K \left[{X}\right]$. Let $\lambda \in K \setminus \left\{{0}\right\}$. Then $P$ is irreducible in $K \left[{X}\right]$ iff $\lambda P$ is also irreducible in $K \left[{X}\right]$. {{expand|Investigate whether this result also holds where $K$ is a general ring.}}
Conjunction of Disjunctions Consequence
:$\left({p \lor q}\right) \land \left({r \lor s}\right) \vdash p \lor r \lor \left({q \land s}\right)$
Existence of Ring of Polynomial Forms in Transcendental over Integral Domain
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$ whose zero is $0_D$. Let $X \in R$ be transcendental over $D$ Then the ring of polynomials $D \sqbrk X$ in $X$ over $D$ exists.
Eisenstein Integers form Integral Domain
The ring of Eisenstein integers $\struct {\Z \sqbrk \omega, +, \times}$ is an integral domain.
Eisenstein Integers form Subring of Complex Numbers
The set of Eisenstein integers $\Z \sqbrk \omega$, under the operations of complex addition and complex multiplication, forms a subring of the set of complex numbers $\C$.
Polynomial Forms over Field form Integral Domain/Formulation 2
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\GF$ be the set of all polynomials over $\struct {F, +, \circ}$ defined as sequences. Let polynomial addition and polynomial multiplication be defined as: :$\forall f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}, g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots} \in \GF$: ::$f \oplus g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$ ::$f \otimes g := \tuple {c_0, c_1, c_2, \ldots}$ where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j \circ b_k$ Then $\struct {\GF, \oplus, \otimes}$ is an integral domain.
Maximal Spectrum of Ring is Nonempty
Let $A$ be a non-trivial commutative ring with unity. Then its maximal spectrum is non-empty: :$\operatorname {Max} \Spec A \ne \O$
Knaster-Tarski Lemma
Let $\left({L, \preceq}\right)$ be a complete lattice. Let $f: L \to L$ be an increasing mapping. Then $f$ has a least fixed point and a greatest fixed point.
Knaster-Tarski Theorem
Let $\left({L, \preceq}\right)$ be a complete lattice. Let $f: L \to L$ be an increasing mapping. Let $F$ be the set (or class) of fixed points of $f$. Then $\left({F, \preceq}\right)$ is a complete lattice.
Degree of Product of Polynomials over Ring/Corollary 2
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $D \sqbrk X$ be the ring of polynomials over $D$ in the indeterminate $X$. For $f \in D \sqbrk X$ let $\map \deg f$ denote the degree of $f$. Then: :$\forall f, g \in D \sqbrk X: \map \deg {f g} = \map \deg f + \map \deg g$
Knaster-Tarski Lemma/Power Set
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $f: \powerset S \to \powerset S$ be a $\subseteq$-increasing mapping. That is, suppose that for all $T, U \in \powerset S$: :$T \subseteq U \implies \map f T \subseteq \map f U$ Then $f$ has a greatest fixed point and a least fixed point.
Cantor-Bernstein-Schröder Theorem/Proof 6
Let $A$ and $B$ be sets. Let $f: A \to B$ and $g: B \to A$ be injections. Then there is a bijection $h: A \to B$; so that $A$ and $B$ are equivalent. Furthermore: : For all $x \in A$ and $y \in B$, if $y = h \left({x}\right)$ then either $y = f \left({x}\right)$ or $x = g \left({y}\right)$.
Ring of Polynomial Forms over Integral Domain is Integral Domain
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {D \sqbrk X, \oplus, \odot}$ be the ring of polynomial forms over $D$ in the indeterminate $X$. Then $\struct {D \sqbrk X, \oplus, \odot}$ is an integral domain.
Union of One-to-Many Relations with Disjoint Images is One-to-Many
Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\mathcal R_1$ be a one-to-many relation on $S_1 \times T_1$. Let $\mathcal R_2$ be a one-to-many relation on $S_2 \times T_2$. Suppose that the images of $\mathcal R_1$ and $\mathcal R_2$ are disjoint. Then $\mathcal R_1 \cup \mathcal R_2$ is a one-to-many relation on $(S_1 \cup S_2) \times (T_1 \cup T_2)$.
Union of Many-to-One Relations with Disjoint Domains is Many-to-One
Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\RR_1$ be a many-to-one relation on $S_1 \times T_1$. Let $\RR_2$ be a many-to-one relation on $S_2 \times T_2$. Suppose that the domains of $\RR_1$ and $\RR_2$ are disjoint. Then $\RR_1 \cup \RR_2$ is a many-to-one relation on $\paren {S_1 \cup S_2} \times \paren {T_1 \cup T_2}$.
Nth Root of Integer is Integer or Irrational
Let $n$ be a natural number. Let $x$ be an integer. If the $n$th root of $x$ is not an integer, it must be irrational.
Union of Bijections with Disjoint Domains and Codomains is Bijection
Let $A$, $B$, $C$, and $D$ be sets or classes. Let $A \cap B = C \cap D = \varnothing$. Let $f: A \to C$ and $g: B \to D$ be bijections. Then $f \cup g: A \cup B \to C \cup D$ is also a bijection.
Interval in Complete Lattice is Complete Lattice
Let $\left({L, \preceq}\right)$ be a complete lattice. Let $a, b \in L$ with $a \preceq b$. Let $\left[{a \,.\,.\, b}\right]$ be the closed interval between $a$ and $b$. {{explain|Demonstrate that for each $a, b \in L$ that $\left[{a \,.\,.\, b}\right]$ exists and is unique.}} Then $\left[{a \,.\,.\, b}\right]$ is also a complete lattice under $\preceq$.
Set Difference with Subset is Superset of Set Difference
Let $A, B, S$ be sets or classes. Suppose that $A \subseteq B$. Then $S \setminus B \subseteq S \setminus A$, where $\setminus$ represents set difference.
Knaster-Tarski Lemma/Corollary
Let $\struct {L, \preceq}$ be a complete lattice. Let $f: L \to L$ be an increasing mapping. Then $f$ has a fixed point
Knaster-Tarski Lemma/Corollary/Power Set
Let $S$ be a set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $f: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a $\subseteq$-increasing mapping. That is, suppose that for all $T, U \in \mathcal P \left({S}\right)$: : $T \subseteq U \implies f \left({T}\right) \subseteq f\left({U}\right)$ Then $f$ has a fixed point.
Alternative Definition of Ordinal in Well-Founded Theory
A set $S$ is an ordinal  $S$ is transitive and $\forall x, y \in S: \left({x \in y \lor x = y \lor y \in x}\right)$.
Foundational Relation is Antireflexive/Corollary
Let $\left({S, \preceq}\right)$ be an ordered set. Suppose that $S$ is non-empty. Then $\preceq$ is not a foundational relation.
Reflexive Reduction of Well-Founded Ordering is Foundational Relation
Let $S$ be a set. Let $\preceq$ be a well-founded ordering of $S$. Let $\prec$ be the reflexive reduction of $\preceq$. Then $\prec$ is a foundational relation.
Epsilon Relation is Proper
Let $\mathbb U$ be the universal class. Let $\Epsilon$ be the epsilon relation. Then $\left({\mathbb U, \Epsilon}\right)$ is a proper relational structure.
Relationship between Transitive Closure Definitions
Let $x$ be a set. Let $a$ be the smallest set such that $x \in a$ and $a$ is transitive. Let $b$ be the smallest set such that $x \subseteq b$ and $b$ is transitive. Then $a = b \cup \set x$.
Ordinal is not Element of Itself
Let $x$ be an ordinal. Then $x \notin x$.
Set is Element of Successor
Let $x$ be a set. Let $x^+$ be the successor of $x$. Then $x \in x^+$.
Element of Ordinal is Ordinal
Let $n$ be an ordinal. Let $m \in n$. Then $m$ is also an ordinal.
Modulus of Exponential of Imaginary Number is One
Let $\cmod z$ denote the modulus of a complex number $z$. Let $e^z$ be the complex exponential of $z$. Let $x$ be wholly real. Then: :$\cmod {e^{i x} } = 1$
Absolute Value of Power
Let $x$, $y$ be real numbers. Let $x^y$, $x$ to the power of $y$, be real. Then: :$\size {x^y} = \size x^y$
Count of Rows of Truth Table
Let $P$ be a WFF of propositional logic. Suppose $\mathcal P$ is of finite size such that it contains $n$ different letters. Then a truth table constructed to express $P$ will contain $2^n$ rows.
Denumerable Class is Set
Let $A$ be a class. Let $\N$ be the natural numbers. Suppose that $F: \N \to A$ is a bijection. Then $A$ is a set.
Relative Complement inverts Subsets
Let $S$ be a set. Let $A \subseteq S, B \subseteq S$ be subsets of $S$. Then: :$A \subseteq B \iff \relcomp S B \subseteq \relcomp S A$ where $\complement_S$ denotes the complement relative to $S$.
Union of Subsets is Subset/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then for all sets $X$: :$\displaystyle \paren {\forall i \in I: S_i \subseteq X} \implies \bigcup_{i \mathop \in I} S_i \subseteq X$ where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the union of $\family {S_i}$.
Union Distributes over Union/Sets of Sets
Let $A$ and $B$ denote sets of sets. Then: :$\displaystyle \bigcup \left({A \cup B}\right) = \left({\bigcup A}\right) \cup \left({\bigcup B}\right)$ where $\displaystyle \bigcup A$ denotes the union of $A$.
Union Distributes over Union/Families of Sets
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Then: :$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$ where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.
Intersection Distributes over Intersection/Sets of Sets
Let $A$ and $B$ denote sets of sets. Then: :$\displaystyle \bigcap \paren {A \cap B} = \paren {\bigcap A} \cap \paren {\bigcap B}$ where $\displaystyle \bigcap A$ denotes the intersection of $A$.
Intersection Distributes over Intersection/Families of Sets
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Then: :$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$ where $\displaystyle \bigcap_{\alpha \mathop \in I} A_i$ denotes the intersection of $\family {A_\alpha}$.
Intersection Distributes over Intersection/General Result
Let $\left\langle{\mathbb S_i}\right\rangle_{i \in I}$ be an $I$-indexed family of sets of sets. Then: :$\displaystyle \bigcap_{i \mathop \in I} \bigcap \mathbb S_i = \bigcap \bigcap_{i \mathop \in I} \mathbb S_i$
Intersection Distributes over Union/Family of Sets
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be a indexed family of subsets of a set $S$. Let $B \subseteq S$. Then: :$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$ where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.
Union Distributes over Intersection/Family of Sets
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$. Let $B \subseteq S$. Then: :$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$ where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.