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For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy." [2]
Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example).
The Mandelbrot set, one of the most famous examples of mathematical visualization. Mathematical phenomena can be understood and explored via visualization. Classically, this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century).
For example, for the function () =, the inverse image of {} would be {,}. Again, if there is no risk of confusion, f − 1 [ B ] {\displaystyle f^{-1}[B]} can be denoted by f − 1 ( B ) , {\displaystyle f^{-1}(B),} and f − 1 {\displaystyle f^{-1}} can also be thought of as a function from the power set of Y {\displaystyle Y} to the power set ...
An example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that fire or combustion is occurring.
The second step of the proof is to show that such solutions always exist, thus concluding the non-existence of a counter-example. This is done using co-dimension 2 mean curvature flow with boundary. The required interior estimates for higher codimensional mean curvature flow in an indefinite geometry appear in [17]. The final part is the ...
In the mathematical field of geometric topology, the Poincaré conjecture (UK: / ˈ p w æ̃ k ær eɪ /, [2] US: / ˌ p w æ̃ k ɑː ˈ r eɪ /, [3] [4] French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave " handedness " unchanged.