**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.

Read more about this topic: Line Segment

### Other articles related to "proof, proofs":

... Mathematicians describe an especially pleasing method of

**proof**as elegant ... Depending on context, this may mean A

**proof**that uses a minimum of additional assumptions or previous results ... A

**proof**that is unusually succinct ...

... However, proving seemingly innocuous statements can require long

**proofs**using only the above eleven axioms ... Consider the following

**proof**that (x + 1)2 = x2 + 2 · x + 1 (x + 1)2 = (x + 1)1 + 1 = (x + 1)1 · (x + 1)1 by 9 ... The length of

**proofs**is not an issue

**proofs**of similar identities to that above for things like (x + y)100 would take a lot of lines, but would really involve little more than the above

**proof**...

... The case n = 0 (no bins at all) allows 0 configurations, unless k = 0 as well (no objects to place), in which there is one configuration (since an empty sum is defined to be 0) ... In fact the binomial coefficient takes these values for n = 0 (since by convention (unlike, which by convention takes value 0 when, which is why the former expression is the one used in the statement of the theorem) ...

... There are many known

**proofs**of the circle packing theorem ... Paul Koebe's original

**proof**is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain ... There are several different topological

**proofs**that are known ...

... of approximation--- and the theory of probabilistically checkable

**proofs**(PCP) and the PCP theorem, which gives stronger characterizations of the class NP ... won the Gödel Prize in theoretical computer science for his papers "Interactive

**Proofs**and the Hardness of Approximating Cliques" and "Probabilistic ...

### Famous quotes containing the word proofs:

“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain *proofs* that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”

—David Hume (1711–1776)

“I do not think that a Physician should be admitted into the College till he could bring *proofs* of his having cured, in his own person, at least four incurable distempers.”

—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)