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• Functions
  ...to any loose collection of objects or entities -- of any nature -- as a ''set''. ...lly, whether the items are arranged in a box or piled up on the counter. A set of encyclopedia consists of books that can be arranged alphabetically or ch
  151 KB (25,679 words) - 17:09, 20 February 2019
• Operations on functions
  ...ack to our the example of five boys that form a set and another set is the set of these four balls: We can form a new set that contains all the elements of the two sets.
  142 KB (23,566 words) - 02:01, 23 February 2019
• Topological spaces
  '''Definition.''' Given an arbitrary set $X$, a collection $\tau$ of subsets of $X$ is called a ''topology<!--\index Of course, instead, we can, and did, start with the set of all neighborhoods $\gamma$ in $X$ and then define open sets as ones wher
  27 KB (4,693 words) - 02:35, 20 June 2019
• Linear algebra
  '''Theorem 1.4.3''' The solution set of a homogeneous linear equation ...''a''₃ = 0, the the equation turn into 0 = 0, and the solution set is '''R³'''.
  27 KB (4,667 words) - 01:07, 19 February 2011
• Linear algebra of Euclidean space
  '''Theorem 1.4.3''' The solution set of a homogeneous linear equation If $a_1=a_2=a_3=0$, the the equation turns into $0 = 0$, and the solution set is ${\bf R}^3$.
  26 KB (3,993 words) - 19:48, 26 August 2011
• Neighborhoods and topologies
  '''Definition.''' Suppose $X$ is any set. A collection $\gamma$ of subsets of $X$ satisfying the properties of: ...r any neighborhoods $U, V\in \gamma$ and any point $x\in U\cap V$ in their intersection, there is a neighborhood $W\in \gamma$ of $x$ such that $W \subset U \cap V
  16 KB (2,758 words) - 00:19, 25 November 2015
• Relative topology
  ...s produce intervals and, conversely, every interval in the $x$-axis is its intersection with a disk: ...$X$ with basis $\gamma$ will acquire its own collection $\gamma _A$ as the set of all of its intersections with the elements of $\gamma$:
  34 KB (6,089 words) - 03:50, 25 November 2015
• Data made Euclidean
  This set is called a ''realization''<!--\index{realization}--> of the graph which is This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or som
  31 KB (5,219 words) - 15:07, 2 April 2016
• Classes of functions
  ...s a ''constant function'', i.e., for some specified element $b$ of $Y$, we set: So, for our set $X$ of boys, we have a special function $G$ from $X$ to $X$ (and another fr
  143 KB (24,052 words) - 13:11, 23 February 2019
• Products
  Generally, for any two sets $X$ and $Y$, their product set is defined as the set of ordered pairs taken from $X$ and $Y$: ...s to demonstrate how we can define the algebraic operations on the product set ${\bf R} \times {\bf R}$ in terms of the operations on either copy of ${\bf
  44 KB (7,951 words) - 02:21, 30 November 2015
• Applications of integral calculus
  ...on of two intervals is the sum of the two lengths minus the lengths of the intersection: ...the union of two regions is the sum of the two areas minus the area of the intersection:
  103 KB (18,460 words) - 01:01, 13 February 2019
• Triangulations
  ...if we don't even have a cell complex but only a topological space, i.e., a set with a collection of open subsets? What topological spaces are polyhedra? ...pen sets have non-empty ''pairwise'' intersections but the only ''triple'' intersection is empty. Why is this important? Notice the pattern:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Quotients
  ...ical issues, let's take care of the underlying ''sets''<!--\index{quotient set}-->. Now, '''notation''' for the quotient set is:
  26 KB (4,538 words) - 23:15, 26 November 2015
• Continuous functions
  This set is defined explicitly and is independent from the existence of the inverse ...appear implicitly in many areas of mathematics. For example, the solution set of the equation $f(x) = b$ is $f^{-1}(\{b\})$. In particular, the kernel of
  42 KB (7,138 words) - 19:08, 28 November 2015
• Bases of neighborhoods
  '''Definition.''' Suppose $X$ is any set. Any collection $\gamma$ of subsets of $X$ satisfying properties: ...r any neighborhoods $U, V\in \gamma$ and any point $x\in U\cap V$ in their intersection, there is a neighborhood $W\in \gamma$ of $x$ such that $W \subset U \cap V
  11 KB (2,025 words) - 14:57, 2 August 2014
• Open and closed sets
  Suppose the set of all [[Neighborhoods and topologies|neighborhoods]] $\gamma$ in $X$ is gi [[Image:open set definition.png|500px|center]]
  4 KB (625 words) - 01:55, 1 October 2013
• Manifolds
  ...dex{locally Euclidean space}--> $n$ if for every $x\in X$ there is an open set $U$ such that $x\in U$ and there is a homeomorphism $h:{\bf R}^n \to U$. Th ...n we would simply consider the standard “local” basis at $a$, which is the set of all open $\epsilon$-balls around $a$:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Functions of several variables
  '''Definition.''' The ''graph'' of a function of one variable $z=f(x)$ is the set of all points in ${\bf R}^{2}$ of the form $(x,f(x))$. ...inition.''' The ''graph'' of a function of two variables $z=f(x,y)$ is the set of all points in ${\bf R}^{3}$ of the form $(x,y,f(x,y))$.
  97 KB (17,654 words) - 13:59, 24 November 2018
• Calculus exercises: part III
  '''Example.''' Suppose $A$ and $B$ are path-connected and $a$ is in the intersection. If $P$ and $Q$ belong to the union, find a path from $P$ to $a$, from $a$ *Set up, but do not solve, a system of linear equations for the following proble
  46 KB (8,035 words) - 13:50, 15 March 2018
• Differentiation
  We set up two functions, for the location and the altitude. Then their composition ...t_1$ your car is the same distance from the intersection as it is past the intersection at time $t_2$. Then the $\Delta S=0$! So, the reading ''can'' be off by a l
  82 KB (14,116 words) - 19:50, 6 December 2018
• Precalculus
  The idea is to use this set-up to produce a correspondence: The idea is to use this set-up to produce a correspondence:
  100 KB (16,148 words) - 20:04, 18 January 2017
• Sequences and their limits
  A function defined on a ray in the set of integers, $\{p,p+1,...\}$, is called an ''infinite sequence'', or simply The last option is used when we treat the sequence as a ''set''.
  64 KB (10,809 words) - 02:11, 23 February 2019
• Classification of points with respect to a subset
  For the background see [[Introduction to point-set topology]]. Given a set $A$ in $X$,
  4 KB (703 words) - 01:55, 1 October 2013
• More about manifolds
  '''Dimension 1:''' When is a level set of $f \colon {\bf R} \rightarrow {\bf R}$ a $0$-manifold? Then we are looking at the intersection of the graph of $f$ and the plane $z=A$, around the point $(a,A)$. Since $f
  9 KB (1,542 words) - 19:58, 21 January 2014
• Applications of differential calculus
  In other words, we find the intersection of the tangent line with the $x$-axis: The equation, which is ''linear'', is easy to solve. The point of intersection is
  59 KB (10,063 words) - 04:59, 21 February 2019
• Vector and complex variables
  ...a line on the plane. Then the solution $(x,y)=(4,2)$ is the point of their intersection: ...equations represent two planes in ${\bf R}^3$. The solution is then their intersection:
  113 KB (18,750 words) - 02:33, 10 December 2018
• From continuity to point-set topology
  #redirect[[Introduction to point-set topology]] Our interest is mainly is algebraic topology. Consequently, point-set topology will be limited to that of the [[euclidean space]], "nice" subsets
  7 KB (1,207 words) - 13:01, 12 August 2015
• Discrete calculus article
  A '''simplicial complex''' <math>\mathcal{K}</math> is a set of [[Simplex|simplices]] that satisfies the following conditions: :2. The non-empty [[Set intersection|intersection]] of any two simplices <math>\sigma_1, \sigma_2 \in \mathcal{K}</math> is a
  27 KB (4,329 words) - 16:02, 1 September 2019
• Several variables
  ...s. The height at the end of the flight is $y_1=0$, so to find the time, we set $y=200-16t^2=0$ and solve for $t$: ...ordinates have make sense. Then, we can choose the domain of $F$ to be the intersection of the domains of $f$ and $g$.
  76 KB (13,017 words) - 20:26, 23 February 2019
• Intersection of a finite collection of open sets is open
  Prove that the [[intersection]] of a finite collection of [[open set]]s is open. ...e intersection of these neighborhoods. It will lie inside the intersection set.
  506 bytes (79 words) - 14:43, 23 February 2011
• Functions in higher dimensions
  *a ''circle'' on the plane is defined to be the set of all point a given distance away from its center; *a ''sphere'' in the space is defined to be the set of all point a given distance away from its center.
  113 KB (19,680 words) - 00:08, 23 February 2019
• Introductory algebraic topology: review exercises
  *[[Can a set to be both open and closed? ]] *[[Intersection of any collection of closed sets is closed ]]
  9 KB (1,553 words) - 20:10, 23 October 2012
• Parametric complexes
  ...persistence and can be removed when the threshold for acceptable noise is set. ...$ defined on a rectangle $R$. Then, given a threshold $r$, its lower level set $f^{-1}((-\infty,r))$ can be thought of as a binary image on $R$. Each blac
  45 KB (7,255 words) - 03:59, 29 November 2015
• A new look at continuity
  Now, an example of an especially “nice” set is ...ion $1$, this is simply a closed interval, yet functions defined on such a set won't fit into the last definition of continuity.
  17 KB (2,946 words) - 04:51, 25 November 2015
• Functions of several variables OLD
  [[Image:path-connected set.jpg|right]] '''Exercise.''' Apply the same proof to show that any [[convex set]] is path-connected (think of a box, a square, a 3d cylinder, etc).
  34 KB (5,636 words) - 23:52, 7 October 2017
• Vector functions
  ...R}^n {\rightarrow} {\bf R}$, then for each $c {\in} {\bf R}$ the ''[[level set]]'' relative to $c$ of $f$ is What would be a level set that is not a curve, but that isn't a maximum or a minimum?
  28 KB (4,769 words) - 19:42, 18 August 2011
• Implicit Function Theorem
  ...\bf R}^n \rightarrow {\bf R}$, then for each $c \in {\bf R}$ the ''[[level set]]'' relative to $c$ of $f$ is What would be a level set that is not a curve, but that isn't a maximum or a minimum?
  2 KB (400 words) - 20:29, 28 August 2011
• Calculus Illustrated -- Notation
  *$x\in X\quad$ “$x$ belongs to set $X$” or “$x$ is an element of $X$”; *$x\not\in X\quad$ “$x$ does not belongs to set $X$” or “$x$ is not an element of $X$”;
  2 KB (438 words) - 22:34, 22 June 2019
• Calculus III -- Fall 2017
  *[http://users.marshall.edu/~saveliev/Teaching/Fall17/math231/set01.pdf Set 1] *[http://users.marshall.edu/~saveliev/Teaching/Fall17/math231/set2.pdf Set 2]
  10 KB (1,596 words) - 13:34, 27 November 2017
• Linear algebra: test 1
  #Prove that the set of all non-zero rational numbers, $\mathbf{Q}\backslash \{0\},$ is closed u #Prove that the intersection of two subspaces is always a subspace.
  4 KB (538 words) - 20:28, 9 September 2011
• Conferences and workshops
  ...rics of polytopes, triangulations, combinatorial fixed point theorems, set intersection theorems, combinatorial convexity, lattice point counting, and tropical geo ...essible to beginning math majors. Such a course should have a very limited set of prerequisites, mostly an appropriate level of "mathematical maturity". I
  8 KB (1,122 words) - 02:52, 24 October 2011
• Vector spaces: introduction
  Because it's not independent from the rest (and we want to make the set of axioms as short as possible). Indeed we derive it: '''Definition:''' A ''vector space'' is a set $V$, where two operations are defined:
  14 KB (2,238 words) - 17:38, 5 September 2011
• More on vector spaces
  Let $V$ be the set of all functions $f \colon {\bf R} \rightarrow {\bf R}$. But, we have to ask: is the set [[closed under operations|closed]] under these operations?
  14 KB (2,471 words) - 21:48, 5 September 2011
• Linear operators: part 4
  *$U={\bf R}^N$ with basis being the set of all vectors and *$W={\bf R}^n$ with basis being the set of all edges.
  13 KB (2,067 words) - 01:11, 12 September 2011
• Nerve of cover
  The open sets have non-empty pairwise intersections but the intersection of all three is empty. So what? Observe: *$n$-simplices corresponding to non-empty intersection of $n$ elements of the cover.
  8 KB (1,389 words) - 13:35, 12 August 2015
• Quotient sets
  i.e., the set of equivalence classes: ...relation $~$ on it, the corresponding ''quotient set'' $X/_{\sim}$ is the set of its equivalence classes:
  3 KB (464 words) - 19:36, 31 October 2012
• More metric complexes
  Furthermore, we connect the centers of the adjacent rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...closed under the operations of complement, countable union, and countable intersection. If, furthermore, $K$ has a volume function, the function $\mu :\Sigma \to
  21 KB (3,445 words) - 13:53, 19 February 2016
• Geometric complexes
  Furthermore, we connect the centers of the adjacent rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...closed under the operations of complement, countable union, and countable intersection. If, furthermore, $K$ has a volume function, the function $\mu :\Sigma \to
  20 KB (3,354 words) - 17:37, 30 November 2015
• Constructions
  We use the following '''notation''' for the quotient set: As we know, it is simply the set of all equivalence classes of this equivalence relation:
  13 KB (2,270 words) - 22:14, 18 February 2016
• Is the intersection of any collection of open sets always open?
  Q: Is the [[intersection]] of any collection of [[open set]]s always open? Under what circumstances ''is'' it open? [[intersection of a finite collection of open sets is open|Answer]].
  306 bytes (45 words) - 09:24, 3 September 2011
• Jordan theorem
  ...rdan theorem.''' ''The [[complement]] of a [[closed curve]] with no self-[[intersection]]s ("simple curve") in the [[plane]] has two [[connected components]].'' ...ion]] of a cubical complex in the plane is a [[open and closed sets|closed set]], hence its [[complement]] is open, so it's not a cubical complex.
  1 KB (200 words) - 09:26, 3 September 2011
• Vector calculus: midterm
  ##The intersection of two linear subspaces is a linear subspace. ##The empty set is a linear subspace.
  4 KB (674 words) - 02:48, 22 August 2011
• Product topology
  ...lead, for any two sets $X$ and $Y$ their [[product set]] is defined as the set of ordered pairs taken from $X$ and $Y$: ...X \times Y$ of $X$ and $Y$ is a topological space defined on the [[product set]] $X \times Y$ with the following [[Neighborhoods and topologies|basis]]:
  8 KB (1,339 words) - 16:53, 27 August 2015
• Introduction to abstract mathematics: course
  **the [[power set]], **[[union]]s, [[intersection]]s, [[complement]]s
  3 KB (373 words) - 16:06, 25 September 2013
• Linear Algebra -- Spring 2013 -- final exam
  ...the main diagonal are equal to zero ($a_{ij}=0$ for $i>j$). Prove that the set of all upper-triangular $n\times n$ matrices form a vector space. #Prove that the intersection of two subspaces is always a subspace.
  2 KB (330 words) - 02:21, 7 May 2013
• Manifolds model a curved universe
  ...''' If every point $a \in C$ has an [[open and closed sets|open]] (in $C$) set $U$, $a \in U$, [[homeomorphic]] to ${\bf R}^1$, then $C$ is called a ''$1$ ...$S$ is a subset of ${\bf R}^N$ such that every point $a \in S$ has an open set $U$ such that $a \in U$ and $U$ is homeomorphic to ${\bf R}^2$.
  10 KB (1,588 words) - 17:11, 27 August 2015
• Examples of maps
  ...nctions|continuous]]. The proof of that below relies on the relevant point-set topology. However, it's unnecessary if only subsets of the Euclidean space As the intersection an open, in $X$, set with $A$, this set is open in $A$. Hence $f_A$ is continuous. $\blacksquare$
  5 KB (918 words) - 16:54, 27 August 2015
• Calculus exercises: part II
  *Set up the Riemann sum for the area of the circle of radius $R$ as the area bet *Suppose the parametric curve is given by \[x=\cos3t,\ y=2\sin t.\] Set up, but do not evaluate, the integrals that represent (a) the arc-length of
  15 KB (2,591 words) - 17:15, 8 March 2018
• Parametric curves
  '''Problem.''' Give the set of all [[affine function]]s passing through $(a,f(a))$ '''Definition.''' Given a set of real numbers $S$, an ''upper bound'' $n$ of $S$ is a number $n < \infty$
  34 KB (5,665 words) - 15:12, 13 November 2012
• Vector calculus: review
  #Give an example of a set $S$ and a point $p\in S$ such that $p$ is a limit point of $S$ and but not ##The intersection of two linear subspaces is a linear subspace.
  7 KB (1,394 words) - 02:36, 22 August 2011
• Homeomorphism
  <center>given $x \in X$, draw a line through $x$ and $N$, find its intersection $y$ with $Y$, then $y = f(x)$.</center> ...ates an [[equivalence relation]]<!--\index{equivalence relation}--> on the set of all [[topological space]]s<!--\index{ topological space}-->.
  13 KB (2,168 words) - 13:09, 7 August 2014
• Homology of cubical complexes
  '''Definition.''' The $k$-th ''homology'' $H_k(K)$ of complex $K$ is the set of all $k$-homology classes of $K$. ...cells as parking lots, $0$-homology as a turn-by-turn instructions from an intersection to another. Solution: [[Image:0-hom and path-con.jpg SOLUTION.jpg|15px]]
  7 KB (1,118 words) - 12:58, 12 August 2015
• Transformations
  ...lines on the plane. Then the solution $(x,y)=(4,2)$ is the point of their intersection: ...he ''integrand'' (and the output is another parametric curve) but a way to set up the ''domain of integration'' (and the output is a number).
  46 KB (7,625 words) - 13:08, 26 February 2018
• Union of a finite collection of closed sets is closed
  Prove that the [[union]] of a finite collection of [[closed set]]s is closed. It follows from the fact that [[intersection of a finite collection of open sets is open]]. Closed sets are [[complement
  364 bytes (60 words) - 13:55, 31 October 2010
• Intersection of any collection of closed sets is closed
  Prove that the [[intersection]] of any collection of [[closed set]]s is closed.
  359 bytes (60 words) - 13:58, 31 October 2010
• Homology and algebra
  '''Definition.''' The $k$-th ''homology'' $H_k(K)$ of complex $K$ is the set of all $k$-homology classes of $K$. ...cells as parking lots, $0$-homology as a turn-by-turn instructions from an intersection to another. Solution: [[Image:0-hom and path-con.jpg SOLUTION.jpg|15px]]
  8 KB (1,386 words) - 18:40, 27 August 2015
• Computational Topology by Edelsbrunner and Harer
  ** [[Convex]] set systems 57 ** [[Intersection theory]] 114
  1 KB (101 words) - 20:23, 5 November 2012
• Basis of topology
  Suppose a set $X$ is given. Any collection $\gamma$ of subsets of $X$ is called a ''basis [[Image:intersection of nbhds.jpg|right]]
  816 bytes (143 words) - 18:41, 2 October 2013