Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

: Every non-empty set A contains an element B which is disjoint from A.

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 element of is a member of the equivalence class.

Every repetition of insertion sort removes an element from the input data, inserting it into the correct position in the already-sorted list, until no input elements remain.

Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms.

Every simple module is cyclic, that is it is generated by one element.

Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.

Every element has a successor ( there is no largest element ).

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

Every time a pixel on a triangle is rendered, the corresponding texel ( or texture element ) in the texture must be found.

Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).

Every rational soul has naturally a good free-will, formed for the choice of what is good.

:" Every group is naturally isomorphic to its opposite group "

Every Shintaido practice begins with warming-up exercises designed to soften and extend the body until it can move naturally, without the tensions of everyday life.

Every Hermitian manifold is a complex manifold which comes naturally equipped with a Hermitian form and an integrable, almost complex structure.

The Commission Report quotes Max Weber, Every bureaucracy seeks to increase the superiority of the professionally informed by keeping their knowledge and intentions secret ... Bureaucracy naturally welcomes a poorly informed and hence a powerless parliament — at least insofar as ignorance somehow agrees with the bureaucracy ’ s interests.

Every polynomial P in x corresponds to a function, ƒ ( x )

Every quantum mechanical observable corresponds to a unique function or distribution on phase space, and vice versa, as specified by Hermann Weyl ( 1927 ) and supplemented by John von Neumann ( 1931 ); Eugene Wigner ( 1932 ); and, in a grand synthesis, by H J Groenewold ( 1946 ).

Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

Every class prepares a float which corresponds with the Homecoming theme or related theme of school spirit as assign by school administrators.

Every infinitesimal region of space time may have its own proper time that corresponds to the gravitational time dilation there, where electromagnetic radiation and matter may be equally affected, since they are made of the same essence ( as shown in many tests involving the famous equation E = mc < sup > 2 </ sup >).

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i. e., a stochastic process

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 cotree T defines a cograph G having the leaves of T as vertices, and in which the subtree rooted at each node of T corresponds to the induced subgraph in G defined by the set of leaves descending from that node:

Every word of length over an-ary alphabet corresponds to a node in this tree at depth.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Every infinite game in which is a Borel subset of Baire space is determined.

# Every infinite subset of X has a complete accumulation point.

# Every infinite subset of A has at least one limit point in A.

* Limit point compact: Every infinite subset has an accumulation point.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

* Every separable metric space is homeomorphic to a subset of the Hilbert cube.

* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

* Every separable metric space is isometric to a subset of the

Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

* Every subset of Baire space or Cantor space is an open set in the usual topology on the space.

* Every arithmetical subset of Cantor space of < sup >( or?

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 ) and second countable.

It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

* Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

* Every finite or cofinite subset of the natural numbers is computable.