Hereditarily finite set

2014-02-07T00:15:36Z

141.225.150.174:

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible. Here are some common dental emergencies: Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached. At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN. Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours. Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary. Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital. Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital. Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief. In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off. If you have any inquiries concerning exactly where and how to use [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist], you can contact us at our own internet site.

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2013-05-02T22:46:27Z

141.225.85.227:

[[Image:Segment definition.svg|thumb|250px|right|The geometric definition of a line segment]]
[[File:Fotothek_df_tg_0003359_Geometrie_%5E_Konstruktion_%5E_Strecke_%5E_Messinstrument.jpg|thumb|historical image – create a line segment (1699)]]
In [[geometry]], a '''line segment''' is a part of a [[line (mathematics)|line]] that is bounded by two distinct end [[Point (geometry)|points]], and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a [[polygon]], the line segment is either an [[edge (geometry)|edge]] (of that polygon) if they are adjacent vertices, or otherwise a [[diagonal]]. When the end points both lie on a [[curve]] such as a [[circle]], a line segment is called a [[chord (geometry)| chord]] (of that curve).

==In real or complex vector spaces==
If ''V'' is a [[vector space]] over <math>\mathbb{R}</math> or <math>\mathbb{C}</math>, and ''L'' is a [[subset]] of ''V'', then ''L'' is a '''line segment''' if ''L'' can be parameterized as
:<math>L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}</math>

for some vectors <math>\mathbf{u}, \mathbf{v} \in V\,\!</math>, in which case the vectors '''u''' and {{nowrap|'''u''' + '''v'''}} are called the end points of ''L''.

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a '''closed line segment''' as above, and an '''open line segment''' as a subset ''L'' that can be parametrized as
:<math> L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}</math>

for some vectors <math>\mathbf{u}, \mathbf{v} \in V\,\!</math>.

Equivalently, a line segment is the [[convex hull]] of two points. Thus, the line segment can be expressed as a [[convex combination]] of the segment's two end points.

In geometry, it is sometimes defined that a point ''B'' is between two other points ''A'' and ''C'', if the distance ''AB'' added to the distance ''BC'' is equal to the distance ''AC''. Thus in <math>\mathbb{R}^2</math> the line segment with endpoints {{nowrap|''A'' {{=}} (''ax'', ''ay'')}} and {{nowrap|''C'' {{=}} (''cx'', ''cy'')}} is the following collection of points:
:<math>\{ (x,y) | \sqrt{(x-c_x)^2 + (y-c_y)^2} + \sqrt{(x-a_x)^2 + (y-a_y)^2} = \sqrt{(c_x-a_x)^2 + (c_y-a_y)^2}\}</math>.

==Properties==
*A line segment is a [[connected set|connected]], [[non-empty]] [[Set (mathematics)|set]].
*If ''V'' is a [[topological vector space]], then a closed line segment is a [[closed set]] in ''V''. However, an open line segment is an [[open subset|open set]] in ''V'' [[if and only if]] ''V'' is one-dimensional.
*More generally than above, the concept of a line segment can be defined in an [[ordered geometry]].

==In proofs==
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a line segment. [[Segment addition postulate|The Segment Addition Postulate]] can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent.

==As a degenerate ellipse==
A line segment can be viewed as a [[Degenerate conic|degenerate case]] of an [[ellipse]] in which the semiminor axis goes to zero, the [[Focus (geometry)|foci]] go to the endpoints, and the eccentricity goes to one. As a degenerate orbit this is a [[Elliptic orbit#Radial elliptic trajectory|radial elliptic trajectory]].

==See also==
*[[Interval (mathematics)]]
*[[Line (geometry)]]
*[[Line segment intersection]], the algorithmic problem of finding intersecting pairs in a collection of line segments
*[[Spirangle]]
*[[Segment addition postulate]]
==References==
*David Hilbert: ''The Foundations of Geometry''. The Open Court Publishing Company 1950, p. 4

==External links==
{{commons|Line segment|Line segment}}
{{Wiktionary|line segment}}
*[http://planetmath.org/encyclopedia/LineSegment.html Line Segment at [[PlanetMath]]]
*[http://www.mathopenref.com/linesegment.html Definition of line segment] With interactive animation
*[http://www.mathopenref.com/constcopysegment.html Copying a line segment with compass and straightedge]
*[http://www.mathopenref.com/constdividesegment.html Dividing a line segment into N equal parts with compass and straightedge] Animated demonstration

{{PlanetMath attribution|id=5783|title=Line segment}}

[[Category:Elementary geometry]]
[[Category:Linear algebra]]

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Hereditarily finite set

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