Note that modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous, though it still makes sense to ask whether subsets are closed.

Note that these properties are also satisfied if " interior ", " subset ", " union ", " contained in ", " largest " and " open " are replaced by " closure ", " superset ", " intersection ", " which contains ", " smallest ", and " closed ", respectively.

Note the dark blue colour and the correct proportions of the union badge in Swedish flags.

Note that in object-oriented programming languages, such as Java, a disjoint union can be expressed by designing class hierarchies.

Note the trade union badge on the right breast.

* Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.

Note that the result here ( also shown dashed ) is not a simple cycle but an edge-disjoint union of two simple cycles.

Note that the union of two transitive relations need not be transitive.

Note that if A and B are disjoint subsets of some weighted set ( U, w ), then the Dirichlet series for their ( disjoint ) union is equal to the sum of their Dirichlet series:

Note: Every cyclic permutation of definition type 3 may be seen as an union of a cyclic permutation of definition type 2 and some fixed points.

Pruning when there's a collarless union: Note there's no swollen area where the branch joins the trunk, this is known as a collarless union, cut at a mirror angle to the BBR ( branch bark ridge )

Note, however, that in both rugby sevens ( usually, but not always, played under union rules ) and rugby league nines, conversions may only take place as drop kicks.

of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. ( Note that although W is not in general a subgroup of G, the coset wB is still well defined.

Note: Delhi is no longer a union territory

Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator.

Note, however, that an incestuous relationship between one or two non-Jews cannot produce a mamzer, and if the product of such a union were to convert he or she would be the equal of any Jew ( Shulhan Arukh Even haEzer 4: 21 ).

Note that the set of ∞- Borel sets may not actually be closed under wellordered union ; see below.

Note that the mass scale is one to two orders of magnitude greater than some previously used ; ;

Note that it is often rational to improve a system in an order that is " non-optimal " in this sense, given that some improvements are more difficult or consuming of development time than others.

( Note that prior to the 1995 renovations, some of these houses, particularly North and Caven, ' traditionally ' sat elsewhere )

: Note: Individual casinos may pay some of these bets at different payout ratios than those listed below.

Note: it is reported in the scientific literature that some " shock " features, such as small shatter cones, which are often reported as being associated only with impact events, have been found in terrestrial volcanic ejecta.

Note that outside of the US military and some US government agencies, including the Internal Revenue Service, this format is usually referred to as " ordinal date ", rather than " Julian date "

Note: There is some confusion about the term ' compression ratio ', particularly outside academia and commerce.

Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angle s. The latter sort of properties are called invariant ( mathematics ) | invariant s and studying them is the essence of geometry.

Note: The all or none principle is met when all patients died before the Rx became available, but some now survive on it ; or when some patients died before the Rx became available, but none now die on it.

Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides.

Note that not all worlds have the same objects: some objects exist in some worlds and not in others.

Note that a grid network is not the same as a grid computer ( or computational grid ) ( even though the nodes in a grid network are usually computers, and grid computing obviously requires some kind of computer network to interconnect the computers ).

The Village Gate, the Village Vanguard and The Blue Note ( since 1981 ), hosted some of the biggest names in jazz on a regular basis.

( Note: Revenue collected from Karachi includes revenue from some other areas since the Large Tax Unit ( LTU ) Karachi and Regional Tax Offices ( RTOs ) Karachi, Hyderabad, Sukkur & Quetta cover the entire province of Sindh and Balochistan ).

Note that these arguments may indeed have validity, but they require some independent justification of the connection between their terms: otherwise the argument ( as a logical tool ) remains fallacious.

Note: These totals include some foreign-owned ships registered here as a flag of convenience: Denmark 13 ( 2002 est.

Note: The name of the book appears next to some author references.

Note that some metres have an overall rhythmic pattern to the line that cannot easily be described using feet.

Note that some of these tasks have direct real-world applications, while others more commonly serve as subtasks that are used to aid in solving larger tasks.

Note that some languages have more such ambiguity than others.

Note that most of these verses do not distinguish between the conditional election ( Arminian ) and unconditional election ( Calvinist ), but are simply evidence of some type of election.

After some time spent inactive, he was asked to play the famous Half Note in New York City in 1971 by guitarist Jim Hall.

Note that in many concrete contexts, one would refer to the object K as the " kernel ", rather than the morphism k.

Note: in non-medical contexts, the word " fistula " is often used to imply urogenital fistula

Note that the stère as used in contexts outside the timber industry is not subject to the same ambiguity.

Note that the procedure is called secondary, not the resulting PDP contexts that have no such relationship with the one the PDP address of which they reused.

( Note that " Helsinki Metropolitan Area " and the other terms used are not firmly established and may vary in different contexts.

Note: A number of modern scholars, working on various cultures, have gathered material showing that in some cases the idea of aniconism in religion is an intellectual construction, suiting specific intents and historical contexts, rather than a fact of the tangible reality ( Huntington for Buddhism, Clément for Islam and Bland for Judaism — references in the appropriate follow-up links ).