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. M

    Intersection with empty index set

    If I is empty, and the collection of sets {A_i} is indexed by I, then the intersection of all the A_i is equal to the universal set. Can someone explain why? Or better yet, give a proof?
  2. S

    Trouble finding curve intersection

    [SOLVED] trouble finding curve intersection Homework Statement Sketch the area of the region bounded by the curves y^2+4x=0 and y=2x+4. Set up two integrals, one with respect to x and one with respect to y, for finding the area of the region. Evaluate one of the integrals to find the area...
  3. S

    Calc III: Vector parallel to the line of intersection

    Homework Statement Hi there, Two points on the sphere S of radius 1 have spherical coordinates P: \phi = 0.35 \pi, \theta = 0.8 \pi and Q: \phi = 0.7 \pi, \theta = 0.75 \pi. Find a vector parallel to the line of intersection of the tangent planes to S at the points P and Q. 2. The...
  4. C

    How Can I Prove the Equality of Union and Intersection in This Homework?

    Homework Statement Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A) Homework Equations The Attempt at a Solution Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off...
  5. C

    Prove Intersection of A and B is Empty Set if B Subset of X/A

    Homework Statement Prove A intersects B=empty set if and only if B is a subset of (X/A) Homework Equations The Attempt at a Solution Would I prove the contrapositive in this case? If B is not a subset of (X/A), then the intersection of A at B is not the empty set Could...
  6. M

    Intersection of Two Planes in R3: Always a Line?

    Homework Statement True or False: The intersection of two planes in R3 is always a line. The Attempt at a Solution I'm pretty sure that this statement is true because two planes can only be parallel, or they must intersect in a line because the are infinate. But I have no ideas on...
  7. R

    Curve of Intersection in two Three-Dimensional EQs

    Determine the curve of intersection of the surfaces x^2 + y^2 + z^2 = 4 and x + y + z = 1. The curve should be in parametric form. With this problem, I'm really not sure what direction to go in. I had thought about using the quadratic formula in some manner, but really unclear. Any advice...
  8. Peeter

    Geometric algebra solution for intersection of planes in RN

    Given the parametric representation of two planes, through points P and Q respectively x = P + \alpha u + \beta v y = Q + a w + b z Or, alternately, with u \wedge v = A, and w \wedge z = B x \wedge A = P \wedge A y \wedge B = Q \wedge B It's easy enough to find...
  9. B

    Parametric equation of the intersection between surfaces

    [SOLVED] Parametric equation of the intersection between surfaces Homework Statement Given the following surfaces: S: z = x^2 + y^2 T: z = 1 - y^2 Find a parametric equation of the curve representing the intersection of S and T. Homework Equations N/A The Attempt at a Solution The...
  10. G

    Union and Intersection of sets

    [SOLVED] Union and Intersection of sets Homework Statement a) Find: \bigcup_{i=1}^{\infty}} A_i b) Find: \bigcap_{i=1}^{\infty}} A_i Where A_i = (0,i), that is, the set of real numbers x with 0 < x < i I was doing okay when they gave me A_i = {i, i+1, i+2, ...}, but now that...
  11. G

    Union and Intersection of sets

    Homework Statement a) Find: \bigcup_{i=1}^{\infty}} A_ib) Find: \bigcap_{i=1}^{\infty}} A_i Where A_i = (0,i), that is, the set of real numbers x with 0 < x < i I was doing okay when they gave me A_i = {i, i+1, i+2, ...}, but now that they're giving me (0,i), and introducing x, I'm getting...
  12. R

    Proving an intersection empty set

    I'm trying to prove \bigcap^{\infty}_{n=1}(0,1/n) = EMPTY SET. One thing that I can't seem to bypass is getting past the closed intervals as stated in the Nested Interval Property, which states "For each n\in N, assume we are given a closed interval I_n = [a_n, b_n] = {x \in R : a_n \leq x...
  13. H

    Finding the Point of Intersection: A Simpler Approach

    The question asks that I : Find the point of intersection for each pair of lines. a) x + y= 4, x - 2y=1 b) x + 2y= 0, x - y= 3 c) 2x + y= 1, x + y= 2 d) 6x= 12 - 3y, 1/2y - x= -5 e) 1/2x - y=8, x + 1/3y= 2 f) 5 + y= 4x, x + 2= 2/3y I understand the formula of y=mx+b, m being the...
  14. J

    What acceleration will bring you to rest right at the intersection?

    You are driving to the grocery store at 20 m/s. You are 110m from an intersection when the traffic light turns red. Assume that your reaction time is 0.50s and that your car brakes with constant acceleration. How far are you from the intersection when you begin to apply the brakes? What...
  15. W

    What Does the Bracket Operation < , > Signify in the Intersection Form Qm?

    Hi, everyone: I am trying to understand the intersection form, and I am having trouble with the notation used in Wikipedia's entry on intersection theory: http://en.wikipedia.org/wiki/Intersection_theory_%28mathematics%29 Now, I am somewhat weak in my cohomology, and I...
  16. I

    Limits, shrinking circles, points of intersection and love

    Homework Statement http://www.nellilevental.com/calc1.jpg Homework Equations C1 = \left(x - 1\right) ^{2} + y^{2} = 1 C2 = x^{2} + y^{2} = r^{2} The Attempt at a Solution Initially I thought that the "value" of R was just going to increase without bound since the slope of the line was...
  17. D

    Factoring f(x)=x^3+x to Find x-Axis Intersection

    Ive got a function of f(x)=x^3+x and I need to use factors to show that the graph crosses the x-axis once only. I just factorised it to x(x^2+1) which isn't very helpful, and if i divide everything by x and complete the square with x^2+1 i get a negative number and stuff... any help?
  18. P

    Subrings of a Ring: Intersection as Subring

    Homework Statement Show that the intersection of any two subrings of a ring is a subring. The Attempt at a Solution It seems abstract. suppose a+b=c and a*b=d Then if c is in A and B (where A and B are subrings) then the intersection of A and B denoted by C contains c and if C...
  19. A

    How Do I Find the Intersection of the Surfaces z=x^2+y^2 and x^2+y^2+z^2=2?

    z=x^2+y^2 and x^2+y^2+z^2=2...I need to find the intersection of these two surfaces. Would I just substitute z=x^2+y^2 into the equation of the sphere to find the curve of intersection? But when I do that I get an equation with fourth powers and I don't know what kind of curve that makes.
  20. K

    Curve of Intersection: Finding the Intersection Between a Sphere and an Ellipse

    Homework Statement I am asked to find the curve of intersection between x^2 + y^2 + z^2 = 36 and 2x + y -z = 2. Homework Equations The Attempt at a Solution I know the first equation is a sphere of radius 6, and the second equation will pass through it, so when projected on the xy...
  21. F

    Intersection of 2 curves

    I'm trying to find where x+y=1 meets x=2(y^2) To solve for y I set up: 2(y^2)=1-y 2(y^2)+y=1 y(2y+1)=1 I have y=1 and 2y+1=1 for 2y+1=1, 2y=0 so y=0 y=0,1 But, I notice that my teacher did: 2(y^2)+y-1=0 (2y+1)(y-1)=0 y=-1, 1/2 Why are these 2 methods bringing about different answers...
  22. M

    Finding the point of intersection of two lines

    [SOLVED]Finding the point of intersection of two lines Hi, I would really, really appreciate it if someone could help me with this. Homework Statement Find the point of intersection between the lines: R_1(\lambda)=[1,\hspace {4} 0 \hspace {4} ,-1] + \lambda[1, \hspace {4} 1,\hspace {4} 1]...
  23. Z

    Find the equivalent intersection point of multi lines in 3D space

    Find the equivalent intersection point of multi lines in 3D space. HI everyone, I'm not a native english speaker, so I wonder you could understand my question very well. This question originates from my physics experiments. When I catch several lights from my equipment, the light...
  24. S

    Calculating the Intersection of Two Cylinders at Right Angle

    Homework Statement Consider two quarters of a cylinder of radius R. What is the volume of a the intersection of a two quarters of two cylinders if the quarters meet at a right angle? Two sides of the bounded region will be rectangular according to this arrangement. (see the diagram when it...
  25. Z

    Is the Intersection of Subspaces a Subspace in a Vector Space?

    Hi to all, I really need help fast. How do I solve this question? A solution would be much appreciated. THANKS A MILLION! ======================================================= Let S1 and S2 be the two subspaces in a vector space V. Show that the intersection of S1 and S2 is also...
  26. P

    Finding the Intersection of Two Planes using Vector Equations

    Homework Statement Two planes r_1 and r_2 have the equations: r_1 = ( 1 - \lambda ) \underline{i} + ( 2 \lambda + \mu ) \underline{j} + ( \mu - 1 ) \underline{k} r_2 = ( s - t ) \underline{i} + ( 2s - 3 ) \underline{j} + ( t ) \underline{k} If a point lies in both r_1 and r_2 then...
  27. M

    Finding the Intersection of Graphs for Homework Equation

    Homework Statement 1 = absolute value( (sin(x)-x)/(sin(x))) * 100 Homework Equations Don't think there are any. The Attempt at a Solution I decided to do it graphically, so i graphed y=.01 and y= absolute value(1-(x/sin(x)) I got x = .244 radians for the intersection and I am just...
  28. O

    Intersection of sets spanned by polynomials

    Let s1 be the set spanned by the polynomials: x^3+x+1, x^3-3x^2+x-2, 2x^3-1. Let s2 be the set spanned by the polynomials: x^3-1, x^2+x+1. What is the intersection of s1 and s2? I really don't know where to begin, I don't know how to define these sets, s1 and s2. since i don't know what...
  29. D

    Intersection of Two Coplanar Lines in 3 Dimensions

    Homework Statement Two lines in space are in the same plane. Line AB passes through points A(x,y,z) and B(x,y,z), and line CD passes through points C(x,y,z) and D(x,y,z). Determine if these two lines are parallel. If they are not, determine the x,y,z coordinates where these two lines...
  30. M

    Finite-Compliment Topology and intersection of interior

    [SOLVED]Finite-Compliment Topology and intersection of interior Homework Statement Given topological space (R^{1}, finite compliment topology), find counter example to show that Arbitary Intersection of (interior of subset of R^{1}) is not equal to Interior of (arbitary intersection of...
  31. S

    Velocity of Motorist with Respect to Police Car at Intersection

    Homework Statement Two highways intersect. A police car P is 800 m west from the intersection and moving at 80km/h west. Motorist M is 600m north of the intersection and moving at 60 km/h south. a) in the unit-vector notation, what is the velocity of the motorist with respect to the police...
  32. K

    : Ball intersection problem

    1. A ball is thrown upward from the ground with an initial speed of 25m/s. At the same instant, a ball is dropped from rest from a building 15m high. After how long with the balls be at the same height? 2. The equation i had in mind was D=Vi(t) + 1/2 at^2 3. For this problem I believe that I...
  33. T

    Determining 4 intersection points of two polar graphs?

    Homework Statement r = 2 r^2 = 9sin(2theta) Find the 4 points of intersection Homework Equations The Attempt at a Solution Since r = 2, 4 = 9sin(2theta)... 4/9 = sin(2theta) Taking the inverse of 4/9 only gives me one answer on the calculator (obviously), and I do not know...
  34. T

    Is the intersection of two open sets always open?

    [SOLVED] intersection of two open sets does anyone know how to prove the intersection of two open sets is open?
  35. E

    Is Every Set in a Metric Space an Intersection of Open Sets?

    Homework Statement Prove that any set in a metric space is an intersection of open sets. Homework Equations The Attempt at a Solution I think the general idea would be that this set is in a (probably infinite) number of open sets, so we just take the smallest one. But I am not...
  36. Loren Booda

    Intersection of concentric, unit area sphere and square

    Where do a concentric sphere and square, both of area 1, intersect?
  37. K

    Breaking into traffic at an intersection

    I was given a very open ended question stating... "How Long will it take to break into a line of traffic at a 'T' intersection" I need some sort of start like a theory that includes this kind of thing. We have been studying probability this term so i need something to do with a probabilty...
  38. T

    Intersection points of a line on a circle

    Hi all, I have a bit of an algebraic problem, and my lack of attention during math is starting to show. I was experimenting with hit detection based on lines for a simple shooting game, and the enemies bieng circulair. I already have a function to check the distance from the line to the...
  39. P

    Finding points of intersection

    the parabola equation is: (x^2/25) - (y^2/9) = 1 the line is y = 4-x according to my calculations, if i point y - 4-x into the equation, i get -16x^2 + 200x - 175 = 0. is that right so far? ~Amy
  40. P

    Finding Points of Intersection of a Parabolic Arch and a Hill

    "a parabolic arch has an equation x^2 + 10y - 10 = 0. the arch is on a hill with equation y = 0.1x-1. (measurements are in metres). "find the points of intersection" for this i substituted y =0.1x - 1 into the equation x^2 + 10y - 10 = 0: x^2 + 10(0.1x - 1) - 10 = 0 x^2 + x - 10 - 10 = 0...
  41. edward

    Pan Handler Working an Intersection

    What I did today, by edward Outside of the city limits of Tucson panhandlers can stand in the center median and wave their "will work for food" signs. I was stopped at an intersection so wide that a pedestrian could not be expected to make it all the way across on the walk light. It had a...
  42. S

    Finding Parametric Equations for Tangent Line of Surface Intersection

    Homework Statement 5. Find parametric equations for the tangent line to the curve of intersection of the surfaces z^2 = x^2 + y^2 and x^2 + 2y^2 + z^2 = 66 at the point (3, 4, 5). The Attempt at a Solution f(x,y,z) = x^2 + y^2 - z^2 g(x,y,z) = x^2 + 2y^2 + z^2 Partial derivz...
  43. Z

    Solving 2 Line Equations: Find Point of Intersection

    Homework Statement One of those annoying questions that should be simple, but that I've forgotten how to do: Two lines are given by the equations r1=a+lp and r2=b+mq. Find the condition for the lines to cross, and find there position of intersection. Homework Equations The Attempt...
  44. L

    How to determine the point of intersection of sine and cosine?

    Homework Statement Im not sure how to start this question: determine the points of intersection between y=sin x and y=cos 2 x for x between 0 and pi. The Attempt at a Solution First thing that comes to mind is the eqaute the two, but i don't know how that helps me?
  45. A

    Intersection between a plane and a line

    Hi all: Given a plane ax+by+cz+d = 0, and a straight line, X = X0 + vt. What is an efficient way to compute the intersection point please? Also, is there any efficient method to determine if two points are located on the same side of the plane or on the different side of the plane...
  46. T

    Are these two lines intersecting?

    Hi; First i you read my problem and feel that it does not belong to this forum please inform me which one is the right one. I have 2 lines L1,L2 in 3-dimensions L1 has (x1,y1,z1) (x2,y2,z2) L2 has (x3,y3,z3) (x4,y4,z4) How can I know that these two lines are intersected? Thanks toto
  47. D

    Intersection line of two planes

    Hello,so this is prob a v v basic question but I am confused as to how one find the equation of the line of intersection of two planes:e.g. 2x + 3y + 5z = 2 & 4x + 2y + z = 11. is there just one unique solution or will the solution involve parameters??eeks :bugeye:
  48. J

    Exploring the Intersection of Quantum Mechanics & Astrophysics

    Ok, so I need a topic to present. I am taking Quantum mechanics, introduction to astrophysics, and a seminar class... I need a topic for a 30-40 minute presentation for each class, and was wondering if there are any interesting topics that tie QM and astro together... thus, one presentation...
  49. J

    Exploring the Intersection of Quantum Mechanics & Astrophysics

    Ok, so I need a topic to present. I am taking Quantum mechanics, introduction to astrophysics, and a seminar class... I need a topic for a 30-40 minute presentation for each class, and was wondering if there are any interesting topics that tie QM and astro together... thus, one presentation...
  50. R

    Finding Area of Circle and Cardoid Intersection

    Homework Statement Find the area of the region that is inside the circle r = 6cos(theta) but outside the cardoid r = 2 + 2cos(theta) Homework Equations r = 6·cosθ r = 2 + 2·cosθ The Attempt at a Solution intersections of the two curves. 6·cosθ = 2 + 2·cosθ → 4·cosθ = 2...
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