 Page "Field of sets" ¶ 34
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Every and normal Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form. Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form. * Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff. Every time the Flaminica saw a lightning bolt ( Jupiter's distinctive instrument ), she was prohibited from carrying on with her normal routine until she placated the gods. * Every connected graph with only countably many vertices admits a normal spanning tree. Every context-free grammar can be transformed into an equivalent grammar in Greibach normal form. Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies. Every generalized eigenvalue of a normal operator is thus genuine. Every formula in classical logic is equivalent to a formula in prenex normal form. Every first-order formula is logically equivalent ( in classical logic ) to some formula in prenex normal form. Every first-order formula can be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization ( sometimes spelled " Skolemnization "). :* Every generalized eigenvector of a normal matrix is an ordinary eigenvector. Every zombie is restored to normal when Hal Jordan defeats Nekron. Every year the club hosts a two week long ( three weekends ) camp at Bohemian Grove, which is notable for its illustrious guest list and its eclectic Cremation of Care ceremony which mockingly burns " Care " ( the normal woes of life ) with grand pageantry, pyrotechnics and brilliant costumes, all done at the edge of a lake and at the base of a forty-foot ' stone ' owl statue. One particular ad line read " Every normal skin needs these two creams ". Every normal mammalian eye has a scotoma in its field of vision, usually termed its blind spot. Every grammar in Kuroda normal form is monotonic, and therefore, generates a context-sensitive language. Every year thereafter Battle Bowl took place with only one ring and a normal battle royal. Every homeworld ( except Gnasty's World ) contains a flying challenge level where Spyro's normal gliding ability is replaced with the ability to fly freely. Every self-adjoint operator is normal. Every ( nontrivial ) group has a normal series of length 1, namely Every beach soccer match is won by one team, with the game going into three minutes of extra time, followed by a penalty shootout if the score is still on level terms after normal time.

Every and Boolean Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b )( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra. Every Boolean ring R satisfies x ⊕ x Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings. Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )). Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras. Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra. * Every Boolean algebra is a Heyting algebra, with given by. * Every Boolean algebra is a distributive lattice. : Every Boolean algebra contains a prime ideal. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. Every Boolean algebra can be represented as a field of sets. Every finite Boolean algebra can be represented as a whole power set-the power set of its set of atoms ; each element of the Boolean algebra corresponds to the set of atoms below it ( the join of which is the element ). Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. * Every finite Boolean algebra is complete. * Every subset of a complete Boolean algebra has a supremum, by definition ; it follows that every subset also has an infimum ( greatest lower bound ).

Every and algebra Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions. * Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. * Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions. * Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. * Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional. Every division ring is therefore a division algebra over its center. Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ). Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism. Every associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions. Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra. * Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space. Every state on a C *- algebra is of the above type. Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.

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