Every morphism in a concrete category whose underlying function is injective is a monomorphism ; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense.

Every morphism in a concrete category whose underlying function is surjective is an epimorphism.

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 homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.

Every locally constant function from the real numbers R to R is constant by the connectedness of R. But the function f from the rationals Q to R, defined by f ( x ) = 0 for x < π, and f ( x ) = 1 for x > π, is locally constant ( here we use the fact that π is irrational and that therefore the two sets

* Every continuous function f: → R is bounded.

Every empty function is constant, vacuously, since there are no x and y in A for which f ( x ) and f ( y ) are different when A is the empty set.

Theorem Every self-adjoint f in A * can be written as f

Every order-preserving self-map f of a cpo ( P, ⊥) has a least fixpoint.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Every isogeny f: A → B is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary.

Every deterministic complexity class ( DSPACE ( f ( n )), DTIME ( f ( n )) for all f ( n )) is closed under complement, because one can simply add a last step to the algorithm which reverses the answer.

* Every point x of X is isolated in its fiber f < sup >− 1 </ sup >( f ( x )).

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

* Every group G acts on G, i. e. in two natural but essentially different ways:, or.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G

* Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

Every adjunction 〈 F, G, ε, η 〉 extends an equivalence of certain subcategories.

Every adjunction 〈 F, G, ε, η 〉 gives rise to an associated monad 〈 T, η, μ 〉 in the category D. The functor

Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >

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 ).

Water for Every Farm: A practical irrigation plan for every Australian property, K. G.

Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.

Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if

Every year at the International Astronautical Congress, three prestigious awards are given out: the Allan D. Emil Memorial Award, the Franck J. Malina Astronautics Medal and the Luigi G. Napolitano Award.

Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.

Every reduced word is an alternating product of elements of G and elements of H, e. g.

* Every Polish space is homeomorphic to a G < sub > δ </ sub > subspace of the Hilbert cube, and every G < sub > δ </ sub > subspace of the Hilbert cube is Polish.

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

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

* Every representable functor C → Set preserves limits ( but not necessarily colimits ).

Every universal cover p: D → X is regular, with deck transformation group being isomorphic to the fundamental group.

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 functor F: D → E induces a functor F < sup > C </ sup >: D < sup > C </ sup > → E < sup > C </ sup > ( by composition with F ).

# Every subset of an independent set is independent, i. e., for each E ' ⊆ E, E ∈ I → E ' ∈ I.

Every song they wrote was written with an eye toward giving it " deep hidden meaning " or D. H. M.

The last two lines of the Act IV-scene 2 funeral song may also have inspired the lines W. H. Auden, the librettist for Stravinsky's The Rake's Progress, puts into the mouth of Anne Truelove at the end of the opera: " Every wearied body must late or soon return to dust ".

Every adult citizen of this small settlement signed the small petition ; E. K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and R. H. Valentine.

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 H * is very special in structure: it is pure-injective ( also called algebraically compact ), which says more or less that solving equations in H * is relatively straightforward.

Every chain may run in the same direction to form a parallel sheet, every other chain may run in the reverse chemical direction to form an anti parallel sheet, or the chains may be parallel and anti parallel to form a mixed sheet. The pattern of H bonding is different in the parallel and anti parallel configurations.

* Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.

Every module of the form H * is algebraically compact.

Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color ; thus χ < sub > H </ sub >( G ) ≤ | V ( G )|.

Every polytype can have monoclinic ( M ) or hexagonal ( H ) symmetry, and therefore some sources label them, for example, not as 6C, but 6H or 6M depending on the symmetry.

Its publications include Every Stony Acre Has a Name: Celebration of the Townland in Ulster by Tony Canavan, and Townlands in Ulster: Local History Studies, edited by W. H.

She also provided a technical review of the educational text book " The Physics of Every Day Phenomena " by Thomas Griffith ( McGraw-Hill ), “ Physics for the Life Sciences " by Philip Kesten and David Tauk ( W. H.

They recorded the track " Meet Every Situation Head On " together as M. E. S. H ..

Every sitting U. S. President since 1960 ( Eisenhower, John F. Kennedy, Lyndon B. Johnson, Richard Nixon, Gerald Ford, Jimmy Carter, Ronald Reagan, George H. W. Bush, Bill Clinton and George W. Bush ) has addressed a convention or attended an event at Cobo Center.

Every recent president has supported the power of impoundment including Presidents Ronald Reagan, George H. W. Bush, Bill Clinton, George W. Bush, and Barack Obama.

Every complete, connected, simply-connected manifold of constant negative curvature − 1 is isometric to the real hyperbolic space H < sup > n </ sup >.