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. S

    What is the equation for the circle formed by the intersection of two spheres?

    Homework Statement In the space, consider the sphere S1 of radius 3 whose center is the point A (1, -1, 1) and the sphere S2 of radius 2 whose center is the point B(t, 1 - t, 1 + t). a. Find the range of values of t in order the two spheres S1 and S2 have common points b. Find the value of...
  2. S

    Solving Intersection Curve at (1,1,1): Derivatives & Tangent Line

    Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...
  3. P

    Dimension of an intersection between a random subspace and a fixed subspace

    I've been struggling with this problem for about two weeks. My supervisor is also stumped - though the problem is easy to state, I don't know the proper approach to tackle it. I'd appreciate any hints or advice. Let V be an random k-dimensional vector subspace of ℝn, chosen uniformly over...
  4. M

    Product of two subgroups and intersection with p-subgroup

    Let G be a finite group. P is a Sylow p-subgroup of G and K is normal in G also H is a subgroup of G with (|K|,|H|)=1. 1) If p divides |H| then P\capHK is a subgroup of H. 2) Is (1) when K is not normal in G. This is my try of (1); Let y be an element of P\capHK, --> |y| divides...
  5. I

    Intersection of Rationals and (0 to Infinity)?

    Homework Statement Let A = [Q\bigcap(0,\infty)] \bigcup {-1} \bigcup(-3, -2] Homework Equations So A = (0,\infty) \bigcup{-1} \bigcup(-3,-2] The Attempt at a Solution I understand that the Rational numbers are cardinally equivalent to (0,\infty), but why isn't...
  6. L

    Closed set as infinite intersection of open sets

    This is not a homework problem, just something I was thinking about. In a general metric space, is it true that every closed set can be expressed as the intersection of an infinite collection of open sets? I don't really know where to begin. Since the finite intersection of open sets is open...
  7. T

    Linear Algebra: intersection of subspaces

    Homework Statement I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space. Homework Equations M \subset V N \subset V dim(M \cap N) [\vec{v}]_{B_M} is the coordinate representation of a vector v with respect to the...
  8. N

    Whats an infinite intersection of open sets

    whats an infinite intersection of open sets? how is it different from finite intersection of open sets and why is it a closed set in the case of ∞ intersection but open in case of finite. To quote kingwinner, is it being defined as a limit? it really does look look like a limit in the case...
  9. T

    Intersection of planes, curvature and osculating plane

    Homework Statement The equations sin(xyz) = 0 and x + xy + z^3 = 0 define planes in R^3. Find the osculating plane and the curvature of the intersection of the curves at (1, 0, -1)Homework Equations Osculating plane of a curve = {f + s*f' + t*f'' : s, r are reals} Curvature = ||T'|| where T is...
  10. K

    Vector geometry - Intersection of lines

    Homework Statement I have 2 parametric vector equations (of a line) r(t) = (2,-4,4) + t(1,-3,4) s(t) = (1,-1,0) + t(2,-1,1) how do i find the coordinates for which they intersect each other? The answers is (1,-1,0) Homework Equations x=a+λv, for some λ in ℝ (parametric vector...
  11. M

    Finding the intersection points of the two lines in space

    given the lines in space L1 : x = 2t + 1, y = 3t + 2, z = 4t + 3 L2 : x = s + 2, y = 2s + 4, z = -4s – 1 Find the point of intersection of L1 and L2. How do i solve this?
  12. S

    Curve of intersection of surfaces problem (Answer included).

    Homework Statement "Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g, differentiable. (a) What are the derivatives f'(1), g'(1)? (b) What is the...
  13. C

    Need help with conic intersection algebra

    Homework Statement Find intersection points of the following.(Conics are centered to origin) Circle = x^2+y^2=4, Ellipse = (x^2/4)+(y^2/9) = 1 The Attempt at a Solution So far I have this. (BTW I know the solutions are (-2,0) and (2,0) but I'm still unsure how to get there step...
  14. C

    Proof involving intersection

    [b]1. Let A , B and C be sets. Prove that (A-B) ∩ C = (A ∩ C)- B = (A ∩ C) – (B ∩ C). [b]3. Proof: 1st part: Let A, B and C be sets where (A-B) ∩ C. Let X be a particular, but arbitrary element of C. Since C and (A-B) ∩, X € (A-B) and X € C. Therefore, X € A but X ∉ B. Since X is an...
  15. S

    Proving image of intersection?

    Homework Statement Let F be a relation from X to Y and let A and B be subsets of X. Then, F(A \cap B) \subseteq F(A) \cap F(B) The Attempt at a Solution Let y \in F(A \cap B). Then, \exists x \in A \cap B, so \exists x \in A and x \in B. Then, y \in F(A) and y \in F(B), so y \in...
  16. N

    How do you find the intersection of the complements of two neg. dep events?

    Probability ? How do you find the intersection of the complements of two negatively dependent events? I'm given P(a), P(b), and P( A intersect B), but I need to find the conditional probability of 'the complement of B given the complement of A'. I don't know how to find it. I thought I only...
  17. S

    Find a vector tangent to the curve of intersection of two cylinders

    I have attached both the question and the solution. I just have questions as to why the solution is the way it is (sorry if they seem stupid but, while I get how to do it mechanically, I don't understand the fundamental reasoning as to why anything is being done): 1) Why are we taking the...
  18. P

    Complex 3d vector intersection formula

    ok here goes... In a three dimensional environment. i am standing at point (0,0,0) and there is someone else standing at (10,0,0) I start moving with a velocity of (1,2,3)/s and the other guy wants to meet me. I know that he is approaching the point of intersection at 4m/s (that is...
  19. C

    Proving the Empty Intersection of Intervals using Natural Numbers

    Homework Statement Prove that \bigcap_{n=0}^{\inf} (0,\frac{1}{n})=\emptyset The Attempt at a Solution since 0 is not included in our interval. eventually I will get to (0,0) because I could pick a real as close to zero as I wanted and there would be a natural such that...
  20. A

    Ellipse inside a circle - intersection

    Hi everyone. I hope I've found the right place for my first post here. I have a geometry problem which I need to solve for a piece of software I'm writing, and I'm hoping someone might be able to help me. I have a non-rotated ellipse inside a circle, as in this diagram. I know the x and y...
  21. M

    Curve of intersection of 2 functions

    Homework Statement A particle moves along the curve of intersection of shapes y = -x2 and z = x2 in the direction in which x increases. At the instant when the particle is at the point P(1,-1,1), its speed is 9cm/s and that speed is increasing at a rate of 3cm/s2. Find the velocity and...
  22. ElijahRockers

    Angle of intersection between two parametric curves

    Homework Statement This is a problem involving parametric equations. r1= <t,2-t,12+t2> r2= <6-s,s-4,s2> At what point do the curves intersect? Find the angle of intersection, to the nearest degree. The Attempt at a Solution I found the point of intersection, (2,0,16). This is when t=2 and...
  23. T

    Intersection and Addition of Subspaces

    Homework Statement http://img824.imageshack.us/img824/3849/screenshot20120122at124.png The Attempt at a Solution Let S = \left\{ S_1,...,S_n \right\} . If L(S) = V, then T = \left\{ 0 \right\} and we are done because S + T = V. Suppose that L(S) ≠ V. Let B_1 \in T such that B_1 \notin...
  24. A

    Infinite intersection of open sets in C that is closed

    Homework Statement Find an infinite intersection of open sets in C that is closed. The Attempt at a Solution Consider the sets A_n = (-1/n,1/n). Since 0 in A_n for all n, 0 in \bigcap A_{n}. Here I'm a little stuck -- is the proof in R analogous to the proof in C, or do I need a...
  25. A

    Parameterize the intersection of the surfaces

    Parameterize the intersection of the surfaces z=x^2-y^2 and z=x^2+xy-1 What's getting me stuck on this problem is the xy. I set x=t z=x^2-y^2 z=t^2-y^2 z=x^2+xy-1 t^2-y^2=t^2+ty-1 y^2=1-ty Thats as far as of come, I'm stuck on this
  26. T

    Union and Intersection of Sets

    Homework Statement Let A = {x\in R | |x| >1}, B = {x\in R | -2<x<3}. Find A \cup B and A\cap B The Attempt at a SolutionI thought I might attempt this via a number line. Since I don't know how to make a number line in Latex, I'll describe it. I have A as being all of R except for the region...
  27. G

    Finite Intersection of Open Sets Are Always Open?

    Suppose we have non-empty A_{1} and non-empty A_{2} which are both open. By "open" I mean all points of A_{1} and A_{2} are internal points. There is an argument -- which I have seen online and in textbooks -- that A_{1} \cap A_{2} = A is open (assuming A is non-empty) since: 1. For some x...
  28. D

    Dimension of intersection of subspaces proof

    Homework Statement V is a vector space with dimension n, U and W are two subspaces with dimension k and l. prove that if k+l > n then U \cap W has dimension > 0 Homework Equations Grassmann's formula dim(U+W) = dim(U) + dim(W) - dim(U \cap W)The Attempt at a Solution Suppose k+l >n. Suppose...
  29. D

    Sets intersection and the axiom of choice?

    I'm working on some topology in \mathbb{R}^n problem, and I run across this: Given \{F_n\} a family of subsets of \mathbb{R}^n , then if x is a point in the clausure of the union of the family, then x \in \overline{\cup F_n} wich means that for every \delta > 0 one has B(x,\delta) \cap...
  30. V

    Finding orthonormal basis for the intersection of the subspaces

    Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach
  31. B

    Intersection of a sphere and a cone. (projection onto the xy-plane)

    Part of a chapter review problem. Say you have a sphere centered at the origin and of radius 'a'. And you have a (ice-cream) cone which has it's point at the origin and phi equal to ∏/3. How do I find the equation of their intersection? Which is the projection onto the xy plane...
  32. T

    Finding the Volume of the Intersection of Two Cylinders

    Homework Statement Find the volume of the intersection of the two solid cylinders x2 + y2 ≤ 1 and y2 + z2 ≤ 1. The Attempt at a Solution Apparently this is done most easily by cartesian coordinates. I have the integral: \int_{-1} ^1 \int_{-sqrt(1-x^2)} ^{sqrt(1-x^2)}...
  33. A

    Finding intersection between exponential and inverse function

    Homework Statement Given the exponential function and its inverse, where a > 0 Exponential Function: f(x)=a^x Inverse function: f^{-1}(x)=log_ax a) For what values of a do the graphs of f(x)=a^x and f^(-1)(x) intersect?The Attempt at a Solution I have no idea how to start this question. Off...
  34. J

    Intersection of open sets

    Homework Statement a) If U and V are open sets, then the intersection of U and V (written U \cap V) is an open set. b) Is this true for an infinite collection of open sets? Homework Equations Just knowledge about open sets. The Attempt at a Solution a) Let U and V be open...
  35. K

    Intersection of 2 spheres

    Hello, I am calculating some integrals in 3 dimensions. However, the difficulties of such integrals lie in the determination of the boundaries of the variables integrated over. \int_{C} d^{3}\vec{t} e^{-\vec{s}.\vec{t}} For example, if we consider (C) as the region of the intersection of 2...
  36. O

    Abstract algebra , intersection of ideals

    Homework Statement prove that <x^m> intersection <x^n> = <x^LCM(m,n)> Homework Equations The Attempt at a Solution ===> let b be in <x^n> intersection <x^m> then for some t,k,p in Z, b=x^(mt) = x^(nk) thus b=x^(LCM(m,n) * p i.e. b is in <x^LCM(m,n)> <=== let b be...
  37. M

    Sphere and hyperboloid intersection

    Hello all, I'm not sure if there's a simple answer to this, and if there isn't, I won't waste your time; I have so much more to do that I can only devote so much time to this little subproblem. I'm trying to model an eggplant in C++, using a graphics library called Renderman, by way of two...
  38. L

    Finding the tangent line of the curve of intersection

    Given the paraboloid z = 6 - x - x2 -2y2 and the plane x = 1, find curve of intersection and the parametric equations of the tangent line to this curve at point (1,2,-4).So I plugged x=1 into the paraboloid equation and got z = 4-2y2. Then I take the derivative of the curve of intersection...
  39. T

    Finding a subspace (possibly intersection of subspace?)

    Homework Statement Let A be the following 2x2 matrix: s 2s 0 t Find a subspace B of M2x2 where M2x2 = A (+) B Homework Equations A ∩ B = {0} if u and v are in M2x2, then u + v is in M2x2 if u is in M2x2, then cu is in M2x2 The Attempt at a Solution Let B be the...
  40. artfullounger

    Proving that the intersection of any two intervals is an interval

    The question is as follows: Prove that if I1, I2 are intervals and J = I1\capI2 then J is an interval. To be honest I don't even know where to start. There's a "hint" that suggests that I first write out the definitions of I1, I2, J as intervals and of the intersection between I1 and I2, but...
  41. B

    Infinitary union combined with infinitary intersection

    I am struggling with combining infinite unions with infinite intersections, the problem i have is to show that, for Sets Aij where i,j \inN (N=Natural Numbers) ∞...∞ \bigcup ( \bigcap Aij) i=0 j=0 is equal to ...∞ \bigcap{(\bigcupAih(i):h\inNN} ...
  42. J

    Finding Points of Intersection for r = 1 - cos θ and r = 1 + sin θ

    Homework Statement Find all points of intersection of the given curve. Homework Equations r = 1 - cos θ, r = 1 + sin θ The Attempt at a Solution 1 - cos θ = 1 + sin θ 1 = 1 + sin θ + cos θ 0 = sin θ + cos θ After that step, I blank out and can't think about how to get any...
  43. T

    Can Two Planes Meet at a Point Instead of a Line?

    Is it possible for two planes to meet in a point instead of in a line?
  44. J

    Points of Intersection in Polar Areas

    Homework Statement The question is to find the area of the region that lies inside both curves. The part I'm specifically having trouble with is finding the points of intersection. Homework Equations sin (2∅) cos (2∅) The Attempt at a Solution sin 2∅ = cos 2∅ 2 sin ∅ cos ∅ =...
  45. T

    Finding intersection of vector subspaces

    Homework Statement What are the intersections of the following pairs of subspaces? (a) The x-y plane and the y-z plane in R'. (b) The line through (1, 1, 1) and the plane through (1,0, 0) and (0, 1, 1). (c) The zero vector and the whole space R'. (d) The plane S perpendicular to (1, 1...
  46. C

    Finding Intersection of 2 Projectiles: Urgent Help Needed

    (I will try to correctly translate this, as this is a class in french) A person on the board of a swimming pool throws a ball (ball1) at a speed of 8 m / s at an angle of 40 degrees above the horizontal. At the time of launch, the ball is 3 m above the water, and 10 m from the opposite side of...
  47. B

    Line intersection algorithm optimization

    I am trying to heavily optimize a piece of code in C as well as MIPS assembly. Here is a link to my code: http://dl.dropbox.com/u/7264839/P1-3.c http://dl.dropbox.com/u/7264839/P1-4-1%20new.asm The problem is find the number of intersections between 1 pixel wide lines of different colors...
  48. L

    Intersection of paraboloid and normal line

    Homework Statement Where does the normal line to the paraboloid z=x^2+y^2 at the point (1,1,2) intersect the paraboloid a second time? Homework Equations The Attempt at a Solution I found the normal line to be 0=2x+2y-1, but I'm not sure what to do next.
  49. J

    Prove intersection of convex cones is convex.

    1. Let A and B be convex cones in a real vector space V. Show that A\bigcapB and A + B are also convex cones.
  50. A

    Solving the Homework: Parametric Equation and Point of Intersection

    Am I doing this right? Homework Statement A.) Find the parametric equation for the line \overline{L} through (2,-1,4) and perpendicular to the lines: \overline{r_{1}}(t) = <1,2,0> + t<1,-1,3> \overline{r_{2}}(s) = <0,3,4> + s<4,1,-2> B.) Determine the point of intersection of the line...
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