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. R

    Proving Finite Convex Sets Intersection is Convex

    Homework Statement Prove that the intersection of a number of finite convex sets is also a convex set Homework Equations I have a set is convex if there exists x, y in the convex S then f(ax + (1-a)y< af(x) + (1-a)y where 0<a<1The Attempt at a Solution i can prove that f(ax + (1-a)y) <...
  2. F

    Intersection of plane in spherical coordinate system

    Dear Friends, I have below query Available data: Point1 (r1,theta1,phi1) Point2 (r2,theta2,phi2) where in spherical coordinate system r(i)=radius theta(i)=angle phi(i)=azimuth Required output: Line of intersection by individual planes generated by each point i.e. from point1 we...
  3. L

    Horizontal Tangent Lines: Intersection of Cylinder and Plane

    Homework Statement Consider the plane z = x + 2y and the cylinder x^2 + y^2 = 1 (a) Find a vector function r(t) describing their intersection. (b) Find the points if any where the tangent to ~r is horizontal (c) Find an equation for the tangent line to ~r at each of these points.Homework...
  4. Z

    Linear algebra - dimension and intersection

    Homework Statement let V be a finite dimensional vector space of dimension n. For W \leq V define the codimension of W in V to be codim(W) = dim(V) - dim(W). Let W_i, 1 \leq i \leq r be subspaces of V and S = \cap_{i=1}^{r}W_i. Prove: codim(S) \leq \sum_{i=1}^{r} codim(W_i)Homework...
  5. E

    "Intersection Equality iff Function is Injective

    Homework Statement Let A, B be sets, C,D\subset A and f:A\longrightarrow B be a function between them. Then f(C\cap D)=f(C)\cap f(D) if and only if f is injective. Homework Equations Another proposition, that I have proven that for any function f(C\cap D)\subset f(C)\cap f(D), and the...
  6. C

    Finite intersection of closed sets is not necessarily closed

    Hi everyone, I'm reading Rudin's Analysis and in the topology section, he implies that the finite intersection of closed sets is not necessarily closed. (pg. 34) Can someone give an example of this? I can't seem to find one.
  7. D

    Intersection of inequalities problem.

    I need to graph/find numbers for S∩T where S is x^2+y^2 <=100 and T is x+y<=14. I know I can find them simply by choosing/picking them, but are there any other solution ? I thought maybe doing x^2+y^2 <=100 + x+y<=14 = x^2+y^2 + x+y<=14 +100 = x^2+y^2 + x+y<=114 = x^2+y^2 <=...
  8. R

    Can an Injection Prove Equality in the Intersection of Subsets in a Function?

    f: A-->B is a function. A,B are sets. Let A1, A2 be contained in/equal to A. f(A1 intersection A2) is contained in OR equal to f(A1) intersection with f(A2). Show that the equality holds if f is an injection. I know how to prove that it is contained, but not the equal/injection part...
  9. J

    Intersection of Three events (probability)

    Homework Statement Show that for 3 events A, B, C, the probability P of the intersection of A, B, and C is greater than or equal to P(A) + P(B) + P(C) - 2. aka: P(A intersection B intersection C) > or = P(A) + P(B) + P(C) - 2 Homework Equations N/R The Attempt at a Solution Use...
  10. K

    The intersection of a plane and a sphere proof

    Homework Statement When a plane intersects a sphere at more than two points, it is a circle (given). Let x^2+y^2+z^2=1 be a sphere S, and P be a plane that intersects S to make a circle (called C). Let q:[a,b] -> R^3 be a unit speed parameterization whose trace is C. Prove that the second...
  11. atomqwerty

    Determining Intersection Points of Equations in Quantum Mechanics

    Hello, How can I determinate the intersection points of these equations: 1-(2/a)x and +- Exp(-x) It's from an exercise on quantum mechanics where I don't know why the gradient of the exponential is -1. YThanks
  12. S

    How to find the intersection of a cylinder and a plane?

    Homework Statement The plane x+y+z=1 cuts the cylinder x^{2}+y^{2}=1 in an ellipse. Find the points on this ellipse that lie closests to and farthest from the origin.Homework Equations N/AThe Attempt at a Solution first step was to determine the intersection of the plane and the cylinder. so...
  13. I

    How do I find the orthonormal basis for the intersection of subspaces U and V?

    Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg Solution in the image is right but my answer is so off from the current one. Homework Equations The...
  14. Rasalhague

    Cantor's finite intersection principle

    I'm trying to understand the proof given in the last 10 minutes or so of this video lecture, but after some struggle, it occurs to me that I may be misinterpreting what the theorem says. According to this, Cantor's finite intersection principle states the following. Given a metric space (X,d)...
  15. K

    Normal Subgroups intersection = <e>

    Let H and K be normal subgroups of G such that H intersect K=<e>. Show that hk=kh for all h in H and k in K. H and K are normal so ghg^-1 is in H and gkg^-1 is in K. want to show hk=kh. So basically I'm showing this is abelian. Can I do ghg^-1=gkg^-1? ghg^-1g=gkg^-1g gh=gk so that works if g=h
  16. M

    Finding the intersection of subspaces, and addition of subspaces

    Heres the question: Let {u,v,w} be a linearly independent set of vectors of R^4. Let E = span{u,2v} and F=span{w,v}. Find EnF and E + F. i really have no idea other than i guess if 1/2u=w and v=v, then the EnF can be defined by that, but I'm not sure if that is right! :(
  17. W

    What Is the Geometric Difference in Intersection Forms Q(a,b) vs Q(b,a)?

    Hi, everyone: This should be easy, but I am having trouble with it. I am rusty and trying to get back in the game: Let Q(a,b) be an intersection form in the middle homology class of some 2n-manifold. What is the geometric difference between Q(a,b) and Q(b,a).? If n is even...
  18. H

    How to show that a transverse intersection is clean, but not conversely?

    How to show that a transverse intersection is clean, but not conversely?
  19. H

    Finding a tangent vector to the intersection of two surfaces

    Homework Statement The surfaces S1 : z = x2 + y2 and S2 : x2 + y2 = 2x + 2y intersect at a curve gamma . Find a tangent vector to at the point (0, 2, 4). Homework Equations i thought about finding gradients of the two functions and plug in the given point in the gradients and cross...
  20. B

    Interpreting Intersection Form in H_1(K)

    Hi, Everyone: The intersection form q(a,b) in dimension 1 (i.e., in H_1(K) , for any top. space K) is symplectic/alternating , meaning that q(a,b)=-q(b,a). From this last, it follows that q(a,a)=0. How do we interpret this last.?. Does this imply that any curve in any...
  21. I

    Inelastic collision, two cars approach each other at intersection.

    Two cars approach each other at an intersection. One car has a mass of 928.4 kg and is traveling in the negative y direction with a velocity of 21.4 m/s. The second car has a mass of 951.2 kg and is traveling in the positive x direction with a velocity of 39.5 m/s. If the collision is totally...
  22. G

    Using Bisection Method to Find Points of Intersection for y=x^3-2x+1 and y=x^2

    y=x^3-2x+1 y=x^2 The question says to use the bisection method to find the points of intersection of the 2 curves. I know how to use the bisection method to find the root of an equation (like on this page http://kr.cs.ait.ac.th/~radok/math/mat7/step7.htm ), but how would I use it to find...
  23. R

    Can the intersection over a finite set be written as a sum?

    I know the union can be, but how about the intersection? I am trying to prove that: Suppose (X,T) is a finite topological space, n is a positive integer and U_i\in T for 1<= i <= n. Use mathematical induction to prove \bigcap U_i \in T, where the intersection goes from i=1 to n.
  24. H

    Intersection of complex sphere and cone

    Show the intersection of complex sphere (|z1|^2+|z2|^2+|z3|^2=1) in C^3 and the complex cone (z1^2+z2^2+z3^3=1) in C^3 is a smooth submanifold of C^3. I am trying to do it using regular level set, but I am not sure which one of (1,0) or (1,1,0) should be set to be the regular value?
  25. N

    Intersection of planes using cross product

    Hello! I have a quick question regarding the intersection of three planes if the determinant is 0. If there are solutions, there will be an infinite number of solutions. One of the equations for the plane can be ignored as it is a linear combination of the other two, and can be ignored for...
  26. jegues

    Curve of intersection of surfaces

    Homework Statement See first figure attached Homework Equations The Attempt at a Solution I was able to sketch the two curves individually to get an idea of what I'm looking at, but I still can't really visualize how the two curves would intersect each other in the first octant...
  27. nukeman

    * Need help with Finding the point of Intersection - Thanks

    * Need help with "Finding the point of Intersection" - Thanks Homework Statement Can someone please explain how I would solve this: As in find the pair of the given lines point of intersection. L: x - y = 4 M: x + 2y = 7 Now, Do i have to turn these into slope intercept ? I know...
  28. M

    Understanding Plane Intersections in R3

    thought I understood equations of planes in R3 and their intersections, but apparently not. I'm very confused by what seems to be a basic problem: find a vector equation for the line of intersection of x + y + z= 0 and x + z = 0. Is x + z= 0 still a plane even though it doesn't have the...
  29. nukeman

    Pair of lines: \determine point of intersection. Please tell me if I am correct.

    Homework Statement Ok, I think i got it, but can you all tell me if these are the right/proper steps I must do? Determine the point of intersection of the following pair of lines: 3x - 7y = 8 2x + 4y = -12 Now, first step is the use the 2nd equation and turn the 2nd equation into...
  30. Telemachus

    Trajectory as a sufrace intersection

    Homework Statement Well, I must express this trajectory: \vec{r}=(t^2,2t,t^2) as an intersection of two surfaces. I really don't know how to work this. It seems to be some kind of parabola, but I'd really like to see some step by step for solving this. Bye, and thanks off course.
  31. B

    Find Intersection of 3x2 Matrices Using QR Factorization

    Figured it out.
  32. T

    Is finding the intersection of two lines in R4 using the same method as in R3?

    To find the intersection of two lines in R3, you set the lines equal, right? [a,b,c] + d[e,f,g] = [h,i,j] + k[l,m,n] Then split these into three equations, 1. a + d(e) = h + k(l) 2. b + d(f) = i + k(m) 3. c + d(g) = j + k(n) And solve for k and d, correct? If k and d are consistent...
  33. J

    Intersection of surface and plane.

    Parameterizing vector function for intersection of cylinder and plane Homework Statement Problem asks us to find the vector function of the curve which is created when the plane y= 5/2 intersects the ellptic cyl. (x^2)/4 + (z^2)/6 = 5 Homework Equations The Attempt at a...
  34. V

    Line of intersection of two planes

    Hi, I am having difficutly figuring out why the cross product of the normal vectors of each plane gives the direction vector of the line of intersection. Anyone care to try to explain? Thanks!
  35. M

    Vector parameterization of intersection of 2 surfaces

    Homework Statement Find a vector parameterization of the intersection of the surfaces x2+y4+2z3=6 and x=y2 in R3. The Attempt at a Solution I let x=t. Then y3=t I solved the first equation for z in terms of x z = cube root ((t2+(t(cube rt(t)) - 6)/-2) I know this is wrong...
  36. S

    Proving the Intersection of Functions: A Mathematical Study

    Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove your claim.?
  37. H

    Find Point of Intersection of Line & Plane: Where's Superman?

    Homework Statement I need to locate the coordinates if a point of intersection x0,y0,z0 of a plane with equation 2x+y-z=0 and a line that is perpendicular to that plane and passes through a point G(2,1,0). Homework Equations and I understand that this is a normal line and a plane so...
  38. N

    Inverse Images and Sets (union & intersection)

    Homework Statement Suppose f is a function with sets A and B. 1. Show that: I_{f} \left(A \cap B\right) = I_{f} \left(A\right) \cap I_{f} \left(B\right) Inverse Image of F (A intersects B) = Inverse Image of F (A) intersects Inverse Image of B. 2. Show by giving a counter example that...
  39. J

    How to find symmetric equations for the line of intersection of two planes?

    Hi, I have been at this single problem for two hours with nothing to show for it. Find symmetric equations for the line of intersection of the planes. z = 3x - y - 7 z = 4x + 2y - 6 They also give me one of the symmetric equations, z/10. I have over 3 pages of work for this. I...
  40. A

    The hot-air ballon intersection question.

    I am close to positive I am going at this problem the correct way, but there seems to be some error somewhere. This problem is from online homework. Homework Statement A hot-air balloon has just lifted off and is rising at the constant rate of 2.2 m/s. Suddenly one of the passengers...
  41. E

    Line and Elipse intersection

    Compute the intersection of a line and an ellipse centered at (0,0). Ellipse equation is b²x² + a²y² = a²b² where b is the minor axis and a is the major axis. I am having trouble finding A, B, and C- somewhere down the line, I know the Quadratic Eq. is used to find x. I also know AX² + BX +...
  42. C

    Intersection of 2 vectors in 3D, knowing the angle between them

    ** I accidentally posted this in the pre-calc math section first, but I think it's more applicable here...sorry** Homework Statement I need to find the intersection point of two vectors. For vector A, I have it's start point (0,0,0) and it's magnitude in components (-.41, .28, -.08)...
  43. C

    Intersection of 2 vectors in 3D knowing the angle between the two

    Homework Statement I need to find the intersection point of two vectors. For vector A, I have it's start point (0,0,0) and it's magnitude in components (-.41, .28, -.08). For vector B, I only know it's start point (-2.70, -.45, -.21) I also know that the angle between the two...
  44. T

    Finding the Intersection Of 2 Equations (difficult)

    Homework Statement Solve the following equation: e^{2x}=3x^2 Homework Equations The Attempt at a Solution I can find an approximate solution with a graphing calculator easily, but I am interested how you would find the exact solution. I can take the natural log of both sides...
  45. M

    Infinite intersection of open sets

    I understand that the finite intersection of open set is open, but is it true that the infinite intersection of open set is closed? or is it possible for it to be open as well? Thank you, M
  46. A

    Prove: Set of rational numbers cannot be expressed as intersection of open sets

    Homework Statement Show that the set of rational numbers in the interval (0, 1) cannot be expressed as the intersection of a countable collection of open sets. Homework Equations The Attempt at a Solution This sounds like something requiring proof by contradiction. There must be...
  47. B

    Could an object survive passing through two intersecting black holes?

    Would it be possible for something to pass through 2 event horizons,falling into both black holes? also,if it can,what would happen if the objecct in question (lets say a proton) was at a point where the forces acting on it were equal in every direction? say being at the centre of these two...
  48. W

    Interchange Between Union and Intersection: Am I Correct?

    Hello all, I have the following question regarding the interchange between union and intersection. \cup_{q < t} \cap_{s > q} A_{s} = \cap_{s<t} \cup_{q<s} A_{q} = \cup_{q < t} A_{q} Am I correct? Also, can anyone provide me some more resources regarding this kind of interchange in...
  49. T

    Finding Points of Intersection by Substitution

    Homework Statement Find any points of intersection of the graphs by the method of substitution. xy+x-2y+3=0 x^2+4y^2-9=0 Homework Equations The Attempt at a Solution From the second equation I can solve for y: y=\frac{\sqrt{9-x^2}}{2} Plug it into the first equation and...
  50. G

    Intersection of ellipses and equivalent problems

    Does anyone know how to determine whether two ellipses intersect? I don't need the precise points but rather only the answer whether there are points. All my attempts led to 4th order polynomials, which are heavy to solve, but considering that I don't need the actual points I assume there must...
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