What is Intersection: Definition and 711 Discussions

In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the intersection of objects is that which belongs to all of them. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space.
Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with intersection theory.

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  1. Seydlitz

    Proof of Union and Intersection with nothing

    Homework Statement Prove: ##A \cup \varnothing = A## ##A \cap \varnothing = \varnothing## The Attempt at a Solution Intuitively both are true. The first is true because union with nothing will eventually return the original set. The second is true because there is no element that can be in a...
  2. icesalmon

    Minimizing distance from point to intersection of two planes

    Homework Statement identify the point on the line of intersection of the two planes that is nearest to the point (2,1,1) not on this line p1: x + 2y - z - 1 = 0 p2: x + y + z - 3 = 0 Homework Equations The Attempt at a Solution I think I can find the line of intersection by...
  3. FeDeX_LaTeX

    Intersection of Lines (Vectors)

    Homework Statement Show that if the two lines ##\frac{x - c_1}{d_1} = \frac{y - c_2}{d_2} = \frac{z - c_3}{d_3}## and ##\frac{x - d_1}{c_1} = \frac{x - d_2}{c_2} = \frac{x - d_3}{c_3}## intersect, they lie in the plane ##r.(c \times d) = 0## where c = c1i + c2j + c3k and d = d1i +...
  4. M

    Parametrize plane-sphere intersection

    1. Homework Statement . Parametrize a circumference contained in the plane x+y+z=1, centered at (2,-2,1), and of radius 40.2. The attempt at a solution. At first I thought I could intersect the plane x+y+z=1 with the sphere (x-2)^2+(y+2)^2+(z-1)^2=40^2, but then I realized that this is wrong...
  5. A

    Intersection of two independent events

    If A and B are two independent events then P(A intersection B) = P(A).P(B) I don't refute this but it confuses me. What is the sample space in this? For eg: - If A is the event that we get Head while tossing a coin and B is the event that we get 2 while throwing a die, then what will we be the...
  6. R

    Question about vector space intersection properties

    Its been a while since I've done this stuff, and I don't have a text handy. I know that for sets, intersection distributes over union, I don't remember if the same will hold for vector spaces over addition? for example does A \cap (B + C) = A \cap B + A \cap C
  7. S

    Area of intersection between two circles

    Hi, I would very much like someone to help me solve the area of intersection between to intersecting circles (one with the radius r, and one with the radius 1). Tangents at the intersecting point form a 120 degree outer angle. 1. Homework Statement , 2 Relevent equations Here is a...
  8. alyafey22

    MHB Compactness Theorem: Intersection of Compact Sets

    In the Principles of Mathematical analysis by Rudin we have the following theorem If \mathbb{K}_{\alpha} is a collection of compact subsets of a metric space X such that the intersection of every finite sub collection of \mathbb{K}_{\alpha} is nonempty , then \cap\, \mathbb{K}_{\alpha} is...
  9. K

    Velocity of a bee moving along the curve of intersection of 2 surfaces

    Hi, I've attached the problem and the solution. I understand the solution except for one thing. I've circled the part I'm having problems with. How do I decide if the circled part should be f2(x,y,z)=x2-y2-z or f2(x,y,z)=z-x2+y2 I'm sure it has something to do with the fact that the problem...
  10. J

    Finding Parametric equations for the line of intersection of two plane

    Homework Statement Find the parametric equations for the line of intersection of two planes Homework Equations Equations for the two planes... z=x+y,-------(1) 2x-5y-z=1 -----(2) The Attempt at a Solution My answers are not correct so I guess I'm going about it the wrong way. Someone...
  11. W

    Proving Element In Union of Two Infinite Sets Not Necessarily In Intersection

    Problem: Prove that if an element is in the union of two infinite sets then it is not necessarily in their intersection: Proof: Have I solved it correctly?
  12. W

    Proof Writing for Dummies: Intersection & Union Elements

    Problem: Prove that any element in the intersection of two sets is also in their union. I am reading a proof writing book for dummies & the solution given in text is: http://tinypic.com/r/141hn7/5 http://tinypic.com/r/141hn7/5 First Question: In exam/test, is it OK if I write the...
  13. F

    Tracing seismic rays - possible intersection

    This problem is about seismic wave propagation in a non-homogeneous layer over a halfspace. I'm not asking you to solve anything, I've already solved the problem both algebraically and in Matlab. However, the graph that I've gotten mildy surprises me. According to the graph, the seismic rays...
  14. C

    Finding a surface form the intersection of two surfaces- Stokes' Thrm.

    Homework Statement Let \vec{F}=<xy,5z,4y> Use Stokes' Theorem to evaluate \int_c\vec{F}\cdot d\vec{r} where C is the curve of intersection of the parabolic cylinder z=y^2-x and the circular cylinder x^2+y^2=36 Homework Equations Stokes' Theorem, which says that \int_c\vec{F}\cdot...
  15. A

    Intersection of electric field lines.

    Field lines do not intersect because at their point of intersection,we would get two tangents indicating two directions of electric field at that point. Suppose the two filed lines JUST touch at a single point. wouldnt there be only a single tangent at the point??
  16. Y

    Intersection of a sphere and plane

    Homework Statement Show that the circle that is the intersection of the plane x + y + z = 0 and the sphere x2 + y2 + z2 = 1 can be expressed as: x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6) y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6) z(t) = -[2cos(t)]/sqrt(6) Homework Equations The...
  17. U

    The angle of intersection between a curve and a plane

    Homework Statement r(t)=(t^2+t)i+(t^3-4)j+(3-t)k r(t) hits the xy plane at the point (12,23,0). Find the angle on intersection of r(t) with the xy plane at that point. Angle= Homework Equations cosθ=(AxB)/lAllBl The Attempt at a Solution I find the answer was 0.0017 degree...
  18. M

    Index Family Subset Proof: Union subset Intersection

    Homework Statement Let {B_j: j \in J} be an indexed family of sets. Show that \bigcup_{i \in J} B_i \subseteq \bigcap_{j \in J} B_j iff for all i, j, \in J, Bi = Bj. Homework Equations The Attempt at a Solution First show that \bigcup_{i \in J} B_i \subseteq \bigcap_{j...
  19. E

    How to determine all points of intersection in a polar coordinate

    Is there a way to find all points of intersection in a polar co ordinate graph without the need to draw the graph. i/e USing algebra? If so, how?
  20. D

    Points of intersection of Parametric Lines

    Homework Statement I'm told to find the 2 points the two curves P and Q will intersect on and the parametric equations are: P (x=t, y=2t-1) Q (x=3t-t^2, y=t+1) The Attempt at a Solution I know I'm supposed to set x-equations and y-equations equal to each and solve so that...
  21. C

    MHB Length around intersection of polar curves

    Sketch the 2 polar curves r = -6cos(theta), r = 2 - 2cos(theta). a. Find the area of the bounded region that is common to both curves. b. Find the length around the intersection of both curves. I got a, but I don't know what to do for b because in my calculus book it only shows how to find the...
  22. S

    Example: intersection of compact sets which is NOT compact

    Homework Statement Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact. Homework Equations The Attempt at a Solution I'm not even sure what it means for an element of S to be...
  23. G

    Scalar Equation of Plane: x+y+z=4

    Homework Statement Determine the scalar equation of the plane that contains the line of the intersection of the planes x+y+z=4 and y+z=2, if the plane is two units from the origin. Homework Equations direction of intersecting line is M = N1 × N1 The Attempt at a Solution Let y= 0...
  24. Y

    Intersection of Condensed Matter Physics, EE and Technology

    Greetings! I am currently trying to focus my research interests before I begin applying to grad schools this coming fall. When it comes to Physics, I really enjoy Condensed Matter and everything related to it (say, Thermal Physics, Statistical Physics etc). I also took an Electronics...
  25. M

    Earth stereographic projection line intersection

    Hi, Consider you are standing upright and pointing your finger at the ground. Where does the vector coming off the tip of your finger arrive when it hits ground level on the other side of the Earth? ..Think as if you were going to imperviously dig a hole through the Earth and could travel...
  26. N

    Nyquist Plot Intersection with Real Axes

    Hi everyone, I'm real confused and stucked about a point in applying Nyquist stability criterion... now i'll explain why. I know that it's needed to know how many times I'm wrapping the nyquist critical point (-1;0) with my plot, and I'm enough good to draw by hand a nyquist plot, but the...
  27. Patolord

    Double Integral:Finding the Area of an Line Intersection.

    Homework Statement Calculate the area of the figure given by these lines. ;x=y ;x=2y ;x+3y=1 ;x+3y=2 Homework Equations This is the intersection. http://www.wolframalpha.com/input/?i=x%3Dy%3Bx%3D2y+%3Bx%2B3y%3D1+%3Bx%2B3y%3D2...
  28. A

    MHB How to find the intersection between a polytope and a hyperplane

    Good afternoon! I am working on a problem, where I at some point have to find the intersection between a polytope and a hyperplan. Consider the following convex set: x2>=x1>=x4>=x3 x1+x2+x3+x4=C1 where C1 is a number. In matrixform it can be represented in the following way: A1*x<=b1...
  29. L

    A Falling Rod's Intersection Moving Faster Than Light?

    Homework Statement There is a rod falling at a speed v that makes an angle θ with the x-axis as it falls. Is it possible for the intersection point to move faster than light as it falls. Homework Equations The Attempt at a Solution I have done the geometrical calculations and I...
  30. G

    MATLAB Modeling an Intersection with Rules for Traffic Flow in MATLAB

    Hi, I am trying to model an intersection using different rules but I am having a hard time getting the thing to work in the simplest stage. I know there is probably an easier way to do this but I am trying to do it this way: clear grid=zeros(10,10); %grid creation grid2=zeros(10,10)...
  31. D

    Intersection of surface and tangent plane

    Homework Statement I have a surface given by z=x^2 - y^2 and its tangent plane at the point (x,y)=(1,1) given by z = 2x-2y. I am asked to compute the intersection of the tangent plane with the surface. The Attempt at a Solution I did the obvious and set x^2-y^2 = 2x -2y to find the x,y...
  32. C

    Line of intersection of Two Planes

    Homework Statement Please disregard, sign error corrected in the cross product Determiner the line of intersection of the following two planes. Write the parametric equations for this line. 2x+y-2z=5 3x-6y-2z=15 Homework EquationsThe Attempt at a Solution First I crossed my normal vectors...
  33. J

    Finding the point of intersection between two curves. (Vectors)

    Homework Statement At what point do the curves r1(t) = (t, 4-t, 63+t^2) and r2(s)= (9-s, s-5, s^2) intersect? Answer in the form: (x,y,z) = ____ Find the angle of intersection theta to the nearest degree. Homework Equations The Attempt at a Solution i: t=9-s j: 4-t=s-5...
  34. B

    How To Find The Intersection Of Two Lines In Space If At All Possible.

    Homework Statement I am given the equation of two lines that are in three space. They are in the form of (X=, Y=, Z= ). The questions wants me to prove whether or not the lines intersect. Homework Equations The equations of the lines. It gives me just the points in the equation, but it is...
  35. W

    Point of intersection between plane and line?

    intersection between x-2y+3z=11 and <1,0,-2> + t<3,-1,2> my attempted solution (59/11, -16/11, 10/11)
  36. L

    Vector Function of Cone & Plane Intersection Curve

    Homework Statement Find a vector function that represents the curve of intersection of the two surfaces: The cone z = sqrt( x^2 + y^2) and the plane z = 1+y. Homework Equations z = sqrt( x^2 + y^2) and the plane z = 1+y. The Attempt at a Solution This problem can be solved as...
  37. S

    Prove the intersection of two orthogonal subspaces is {0}

    Homework Statement Let A and B be two orthogonal subspaces of an inner product space V. Prove that A\cap B= \{ 0\}. Homework Equations The Attempt at a Solution I broke down my proof into two cases: Let a\in A, b\in B. Case 1: Suppose a=b. Then \left\langle a,b \right\rangle =...
  38. jbunniii

    Countable intersection of F-sigma sets

    My question concerns F_\sigma subsets of \mathbb{R}. An F_\sigma set is one which can be expressed as a countable union of closed sets. I have several books that state that a countable intersection of F_\sigma sets need not be an F_\sigma set (indeed, such sets have their own designation...
  39. H

    Dimension proof of the intersection of 3 subspaces

    Homework Statement Assume V = \mathbb{R}^n where n \geq 3. Suppose that U,W,X are three distinct subspaces of dimension n-1; is it true then that dim(U \cap W \cap X) = n-3? Either give a proof, or find a counterexample.The Attempt at a Solution The question previous to this was showing that...
  40. A

    MHB Two different circles in the plane with nonempty intersection

    Hi. Here is a problem I've been trying to solve for some time now. Maybe you could help me. We have two sets \mathcal {Q} is a set of those circles in the plane such that for any x \in \mathbb{R} there exists a circle O \in \mathcal {Q} which intersects x axis in (x,0).\mathcal {T} is a set of...
  41. S

    MHB Find the intersection value of 3 subsets

    Let a, b and c be three subsets of universe U with the following properties: n(A)= 63, n(B)=91, n(c)=44, The intersection of (A&B)= 25, The intersection of (A&C)=23, The intersection of (C&B)=21, n(A U B U C)= 139. Find the intersection of (A&B&C). I am told the answer is 10. I tried drawing...
  42. Z

    Calculate double integral of the intersection of the ellipse and circle

    how to calculate the double integral of f(x,y) within the intersected area? f(x,y)=a0+a1y+a2x+a3xy The area is the intersection of an ellipse and a circle. Any help will be appreciated, I don't know how to do this. can I use x=racosθ,y=rbsinθ to transformer the ellipse and...
  43. G

    Intersection of two planes in R4

    Homework Statement I have two planes in R4, namely {[2, 0, 0, 1], [1, 1, 2, 0]} and {[-2, 0, 0, 1], [0, 1, -1, 0]}. Homework Equations The Attempt at a Solution Tried to row eliminate, didn't work. Tried figuring out a normal equation, but clearly that won't work in R4. Don't...
  44. C

    Stokes' Thm, intersection of sphere and plane

    Homework Statement Use Stokes' Theorem to evaluate $$\int_{\gamma} y\,dx + z\,dy + x\,dz,$$ where ##\gamma## is the suitably oriented intersection of the surfaces ##x^2 + y^2 + z^2 = a^2## and ## x + y + z = 0## The Attempt at a Solution Stokes' says that this is equal to $$\iint_S...
  45. 7

    Intersection point between n-vector and n-sphere

    Hello, I'm trying to compute the intersection point between a n-dimensional vector and a n-sphere. Do you know how to perform this? is it the same than 2 and 3 dimensions? I can't really find much information about this topic. Thank you very much,
  46. T

    Intersection of a sequence of intervals equals a point

    Intersection of a sequence of intervals equals a point (Analysis) Homework Statement Let A_{n} = [a_{n}, b_{n}] be a sequence of intervals s.t. A_{n}>A_{n+1} and |b_{n}-a_{n}|\rightarrow0. Then \cap^{∞}_{n=1}A_{n}={p} for some p\inR. Homework Equations Monotonic Convergent Theorem If...
  47. T

    Intersection of subgroups is a subgroup

    Homework Statement Suppose H and K are subgroups of G. Prove H intersect K is a subgroup of G. Homework Equations Suppose G is a group and H is a nonempty subset of G. Then H is a subgroup of G iff a,b ∈ H implies ab^-1 ∈ H. The Attempt at a Solution Suppose a and b elements of H intersect...
  48. V

    Showing two epsilon balls intersection is empty

    Homework Statement Suppose x,y \in X which is a normed linear space and x\neq y . Prove that \exists r>0 such that B(x,r) \cap B(y,r)=∅ Homework Equations Epsilon Ball B(x,r)={z \in X:||x-z||<r} The Attempt at a Solution So my attempt here is via contradiction and its not...
  49. camilus

    Product and intersection of ideals of polynomial ring

    Let k[x,y,z,t] be the polynomial ring in four variables and let I=<x,y>, J=<z, x-t> be ideals of the ring. I want to show that IJ=I \cap J and one direction is trivial. But proving I \cap J \subset IJ has stumped me so far. Anyone have any ideas?
  50. P

    Algorithm for testing intersection of point and compound polygon

    I'm trying to find a reasonably fast method for testing whether or not a point (x,y euclidean coordinate system) lies inside a (preferably convex, concave or complex - though different methods for each would be OK) compound polygon with edges consisting of line segments, arcs and/or elliptical...
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