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• Image
  Given a [[function]] $f:X \rightarrow Y$ between two sets, the ''image of subset $A$ of $X$ under $f$'' is the set of all outputs of [[category:sets]]
  312 bytes (59 words) - 16:55, 29 July 2012
• Topology graph
  *The graph represents the hierarchy of the lower and upper level sets of the gray level function. **[[cycles]]: both upper and lower level sets are captured by circular sequences of edges.
  10 KB (1,727 words) - 15:03, 9 October 2010
• Binary images
  ...n of non-overlapping regions, connected sets of black pixels and connected sets of white pixels, that covers the whole image. The partition is achieved by
  10 KB (1,705 words) - 21:26, 18 July 2011
• Linear algebra
  Suppose, A = x + S = y + T (equal as sets), where S, T are linear subspaces. Then, S = T and y ∈ A. Affine subspaces are solution sets of systems of non-homogeneous equations.
  27 KB (4,667 words) - 01:07, 19 February 2011
• Sequences and their limits
  For $S=[0,1]$, any number $M\ge 1$ is its upper bound. However, these sets have no upper bounds: '''Example.''' For the following sets the least upper bound is $M=3$:
  64 KB (10,809 words) - 02:11, 23 February 2019
• Image of function
  Also, given sets $X$ and $Y$, subset $A$ of $X$, and a function $f:X \rightarrow Y$. Then th [[category:sets]]
  249 bytes (50 words) - 03:51, 15 February 2011
• Topology of graphs
  ...tion.''' A ''graph''<!--\index{graph}--> $G =(N,E)$ consists of two finite sets: ...at are known to be path-connected! One can then study the topology of such sets by means of graphs represented as discrete structures:
  25 KB (4,214 words) - 16:08, 28 November 2015
• Calculus Illustrated -- Notation
  *$A \cup B= \{x:\ x\in A\ \texttt{ OR }\ x\in B\}\quad$ the union of sets $A$ and $B$; ...\cap B= \{x:\ x\in A\ \texttt{ AND }\ x\in B\}\quad$ the intersection of sets $A$ and $B$;
  2 KB (438 words) - 22:34, 22 June 2019
• Linear algebra of Euclidean space
  Suppose, $A = x + S = y + T$ (equal as sets), where $S, T$ are linear subspaces. Then, $S = T$ and $y \in A$. Affine subspaces are solution sets of systems of non-homogeneous equations.
  26 KB (3,993 words) - 19:48, 26 August 2011
• Students research projects
  ...I and Cubical Homology'' Abstract: Employing magnetic resonance (MR) data sets, I will investigate the advantages of cubical homology in the examination o ..., we can associate an abstract simplicial complex whose faces are the edge sets of the graphs in the collection. A bounded degree graph complex is a simpli
  11 KB (1,674 words) - 23:20, 25 October 2011
• Implicit Function Theorem
  [[Image:level sets in R2.jpg|right]] [[Image:level sets in R2 not curve.jpg]]
  2 KB (400 words) - 20:29, 28 August 2011
• Triangulations of surfaces
  In terms of sets, every face is a triple, such as $ABC$, and every edge is a double, such as [[Category:Topology]] [[Category:Sets]]
  3 KB (505 words) - 18:17, 27 August 2015
• Calculus of chain maps
  ...\cdot$” stands for the multiplication of real numbers and, as a result, of sets of real numbers, not the (formal) multiples of cells. The same applies to t ...t as above, “$\cdot$” stands for the multiplication of real numbers and of sets of real numbers, not the formal multiplication of cells. The same applies t
  41 KB (7,344 words) - 12:52, 25 July 2016
• Linear operators: part 1
  ...t with the general setup: $f \colon X \rightarrow Y$ [[functions]] between sets. ...\colon X \rightarrow Y$, $g \colon Y \rightarrow Z$, two functions between sets $X,Y,Z$, then their ''[[composition]]'' is
  13 KB (2,086 words) - 19:58, 27 January 2013
• The algebra of cells
  '''Exercise.''' What are the open sets of this space? '''Exercise.''' Explain -- in terms of open, closed sets, etc. -- the topological meaning of adjacency.
  34 KB (5,644 words) - 13:35, 1 December 2015
• Inclusion-exclusion formula
  For sets $A,B \subset X$, [[category:sets]]
  141 bytes (26 words) - 16:53, 6 September 2013
• Vector and complex variables
  ...e two sets are ''almost'' equal to the two axes! If we append $0$ to these sets of eigenvectors, we have the following. For $\lambda = 2$, the set is the $
  113 KB (18,750 words) - 02:33, 10 December 2018
• Can a set to be both open and closed?
  Question: Can a set to be both [[open and closed sets|open and closed]]? Why: Sets aren't like doors...
  276 bytes (42 words) - 06:36, 3 September 2011
• New complexes from old
  ...$ given by $q(x) = [x]$ is a [[chain maps|chain map]]. Then the [[quotient sets|quotient set]] $C(K)/ \sim$ is a chain complex with the boundary operator $ ...ding topological spaces in [[New topological spaces from old]] (also [[New sets from old]]).
  2 KB (310 words) - 13:36, 19 July 2011
• Nerve of cover of sphere
  ...of its vertices to build an open cover on the sphere. Of course, the open sets aren't triangles here but their [[complement]]s. Alternatively, the six hemispheres can serve as the open sets. The resulting simplicial complex is an [[octahedron]].
  763 bytes (118 words) - 12:22, 12 August 2015

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