Intersection Definition and 688 Threads

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...

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...

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...

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?

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...

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...

[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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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

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...

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)}...

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...

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...

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...

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...

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...

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...

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} ...

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...

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

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 ∅ =...

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...

(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...

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.

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.

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...

Homework Statement Let M be a metric space, A a subset of M, x a point in M. Define the metric of x to A by d(x,A) = inf d(x,y), y in A For \epsilon>0, define the sets D(A,\epsilon) = {x in M : d(x,A)<\epsilon} N(A,\epsilon) = {x in M: d(x,A)\leq\epsilon} Show that A is...

What I'm looking for is an algorithm to find the details on the intersection of of a circle and rectangle in two dimensional Euclidean space. The information I need to find is straightforward enough; all I need is to know whether the rectangle and circle are not intersecting, partially...

Homework Statement Find a vector parallel to the line of intersection of the planes given by the equations 2x-3y+5z=2 and 4x+y-3z=7. Homework Equations How do I go about this? I know we have two vectors <2,3,5> and <4,1,-3> but where do I go from here? The Attempt at a Solution...

Homework Statement The line x+y=2a-1 intersects the circle x2+y2=a2+3a-3 at point (m,n). When m*n reaches its minimum value, what is the value of a?Homework Equations Equation One x + y = 2a - 1 Equation Two x2 + y 2 = a 2 + 3a - 3

Hi, All: This is a post from another site that was interesting but was not answered: can I reasonably > argue that three planes in 3-space are not likely > to intersect at a point using the fact that >t GL(3,R); > the subset of invertible 3x3-matrices has measure 0 > in...

I have to decide whether the following is true or false: If A1\supseteqA2\supseteqA3\supseteq...are all sets containing an infinite number of elements, then the intersection of those sets is infinite as well. I think I found a counterexample but I'm not sure the correct notation. I to...

The book I am reading says that \bigcap \phi because every x belongs to A \in \phi(since there is no such A ) , so \bigcap S would have to be the set of all sets. now my question is why every x belongs to A \in \phi.In other word I don't completely understand what this statement mean. sorry if...

Homework Statement The graphs y=abs(x) and y=(x^2)-6 intersect at x=3 and x= -3 What is confusing me is when I set them equal to each other and solve (x^2)-x-6=0 and (x^2)+x-6=0 I get -3,+3,-2,+2 What is the deal with the negative 2 and pos 2? Homework Equations The...

Can someone do it without using a graphing calculator? The question specifically states not to use "Trace". I don't understand how to do it algebraically, and I'd love it if someone could teach me. Please and thanks!

Homework Statement The plane P1 contains the points A,B,C, which have position vectors a=(0,0,0), b=(1,1,8) and c=(0,1,5) respectively. Plane P2 passes through A and is orthogonal to the line BC, whilst plane P3 passes through B an is orthogonal to the line AC. Find the coordinates of r, the...