The intersection of two or more sets, written $A\cap B$ or $\bigcap_{i\in I} A_i$, is the set of all elements contained in *all* given sets.

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Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
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1answer
18 views

Calculating number of students who don't study any language

According to a survey of 100 students, there are 40 students studying English, 30 studying French, and 25 studying Spanish. Inaddition, 8 students are studying English and French, 6 are ...
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32 views

What does it mean for the infinite intersection of nested sets to be empty?

$\bigcap I_i=\emptyset$ where $I_i=(0, \frac{1}{i})$ What does it mean that the infinite intersection is empty?
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20 views

line segment intersection strange results

I'm using this formula. I am getting very strange results with (1,3) to (29,17) and (6,19) to (7,8). I got an X* value of 7. When I plugged this into my intercept calculator it said they intercept at ...
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20 views

Probability Intersections

$A$ and $B$ are two events from certain probability space $\Omega$. Knowing that: $P(A)=0.6$, $P(B)=0.7 $ and $P(A\cup B)-P(A\cap B) = 0.2$ Determine $P(A\cap B)$ It says in this sheet that the ...
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Intersection points of two trignometric equations

I am studying for a SAT II Math2C and I came across this question in Barron's book. Solve $2 \sin(x) + \cos2(x) = 2 \sin^2(x) - 1$ [0<= x <= 2pi] The solution says put the equations in a ...
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25 views

Unit sphere x axis intersections

This is a problem from a vector calculus textbook, Higher Order Derivatives Consider the unit sphere S given by x^2+y^2+z^2=1. S intersects the x-axis at 2 points. Which variables can we solve for at ...
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15 views

How do I define a three space?

I'd like to take a hierarchically-modeled (vertices, lines, faces, etc) 4D object and find its intersection with three-space. In Paul Isaacson's thesis, Computer Graphic Presentation of Hypothesized ...
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84 views

Proving that the set of real numbers is a topological space.

I recently finished an activity provided by a professor where one of the questions was to prove that the set of real numbers is a topological space. The hint provided was to "consider the union of ...
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2answers
12 views

Points of intersection for two polar equations question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between ...
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Volume of intersection between two equal cones with parallel axes

The two infinite cones (nappes) (each 45-degree wide) have parallel axes. They are oriented in opposite directions, and the top of one is inside the other, so that the common volume V is finite. How ...
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Construction of concurrent or parallel lines from a parallelogram (proof by vectors)

I have a problem with this probably easy exercise on vectors. Any help would be great. Let ABCD be a parallelogram. The line parallel to AB intersect BC and AD in points Q and S, respectively. The ...
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25 views

Find points that defines the intersection of an ellipse with a plane.

I want to test for the intersection of two ellipses $E_1$ and $E_2$ in $\mathbb{R}^3$ represented on a computer. In some sense, this isn't a hard problem: ...
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1answer
22 views

Measure on intersections of unions

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...∈\mathcal{A}$, assume that $\sum\limits_{j=1}^{\infty}=\mu (A_j)<\infty$ We have ...
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14 views

Measure of intersections [duplicate]

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...\in \mathcal{A}$, assume that $\sum^{\infty}_{j=1}\mu(A_j)<\infty$ I want to show that ...
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67 views

Prove that the two polynomials intersect each other only at a single point

Here are the polynomials: $$D^K_1(\theta)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-\theta)^{K-i}$$ and $$D^K_2(\theta)=\frac{1}{2}\sum_{i=\lceil{K/2}\rceil}^K ...
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1answer
26 views

Vector Parametrization of a Hyperbolic Paraboloid and a Plane

So I need to find the intersection between a hyperboloid ($z=\frac {y^2}{b^2}-\frac{x^2}{a^2}$) and some related plane ($bx+ay-z=0$). I have tried solving for $z$ and equating the two: ...
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1answer
24 views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
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1answer
30 views

Find the equation of line and finding a point in given example

The outer circle is $x^2+y^2=1$ and the smaller circle is $x^2+(y+1-r)^2=r^2$. The arclength is parameterised anticlockwise with $s=0$ at the bottom as shown. If we know $s_n$ and $s_{n+1}$ can we ...
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1answer
34 views

Longest chord inside the intersection area of three circles

I am currently working on my masters thesis in computer science and I stumbled onto a geometry problem. My goal is to compute the length of the longest possible chord inside the intersection area of ...
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17 views

Intersection problem

I am given an interval $A=(0,1)$ and $B=(1,2)$. How can I show that $A \cap B$ is an empty set? I tried to prove by contradiction by saying an arbitrary element $k\in A\cap B$ so $k \in A$ and $k \in ...
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Finding the limit of sets

I have a problem, where I have difficulties with solving these two exercises. Here is the problem If $C_1$, $C_2$ ... are sets where $C_k \supset C_{k+1}$, where k=1,2,3. Then the limit of $C_k$, ...
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1answer
42 views

How to compute coordinates of a point that intersects an sphere

Hi all. Is there a way to compute the S(x,y,z), given the following information: A(x,y,z) e = elevation (from the line AS) Az = azimuth (over A). Perpendicular to x axis. Can vary from 0 tp 360. ...
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22 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
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How to find the point of intersection with three equations?

Given the following equations with three variables $a, b, c$ $a-5b+4c=-3$ $2a-7b+3c=-2$ $-2a+b+7c=-1$ How can I determine the point (if it exists) at which all three lines intersect?
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The intersection of $3$ set is empty, would the intersection of $4$ sets be empty?

Let me clarify some more. Let's say we have four sets $A,B,C,$ and $D$. If the intersection of any three sets is empty, by default is the intersection of all four sets empty?
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43 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
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1answer
13 views

calculate the coordinates of the intersection between a bisector and a sector

I have a sector and would like a formula that gives the intersection between the bisector and the arc. here's a graph of the situation: the point B is the center of a circle of radius AB and the BD ...
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45 views

Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
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36 views

length of secant line.

I'm looking for way to find the length of a secant line intersecting another line through the center of a circle with a known radius. The intersection point is on the circle and the angle between 2 ...
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61 views

Circle Line segment intersection

I have a circle with radius r and center $(c_x, c_y)$. I have a line segment $(x_1, y_1)$ and $(x_2, y_2)$ given $(x_2, y_2)$ is always a point inside the circle. I am trying to find the ...
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1answer
48 views

Union of Intersections [closed]

What is the general formula for finding the union of intersections? For example , Consider the sets $A , B , C $ I need to find $$((A \cap B)\cup(A \cap C)\cup(B \cap C))$$ Please extend this as a ...
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1answer
45 views

Projective and affine conic classification

I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is "Multiple View Geometry in Computer Vision", which, as the title ...
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1answer
36 views

Outer interval of circle intersection

Is there a consistent way to calculate the outer interval $\left(~\mbox{element of}\ \left[0, 2\pi\right]~\right)$ of a circle created by an intersection ?. I calculated the intersection points and ...
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49 views

calculate circle segment area: determine distance

I have a problem calculating the area of a circle segment. I know how to separate this into smaller tasks (triangle and remaining circle segment) that are basically easily solvable, but one distance ...
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Testing hypothesis about window non-overlap

I have a large number (~1.5 million) of protein sequences, each of them of different lengths.There are 6 schematic examples in the attached image. Within each of these sequences, there are >= 0 ...
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38 views

Intersection between 2 lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point (9, -9, 21) I tried solving this myself, I got x = x, y = y, but I could not find a point where ...
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intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
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1answer
55 views

Determining intersecting points between square and circle

I unfortunately have spent too much time trying to solve this question, and have turned to you for help. The corner of my square has intersected some circle, and I need to move it out. I only know one ...
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2answers
63 views

Geometry : find the points of tangency between two lines and two circles [closed]

I have a programming problem. I need to find the intersection points between two lines tangent to two circles and the circles! I have the circles' radiuses and centers. So I need points ...
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2answers
58 views

May a monoid have two disjoint submonoids?

I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any ...
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1answer
111 views

Check if point lies on a line segment

I know there are shorter solutions that use dot product, but I don't know what the logic behind doing so involves so I came up with something that I understand myself (i will research the dot product ...
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2answers
128 views

Parametric equation for intersection of curve

Here's the three part question: A) Find parametric equations for curve which is the intersection of the cylinder $x^2 + z^2 = 1$ and the plane y = -x. B) Show that the curve lies on the surface $x^2 ...
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3answers
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Euclidean “straight line” calculation

Please see image first.. I have as input the following (I presume these are in effect Euclidean coordinates): The angle and the length of the red line. The angle and the length of the green ...
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77 views

How to determine if two ellipse have at least one intersection point

All of the question are in sequence and related. 1.Given 2 ellipse with the position x1,y1, x2,y2 and the radius a1,b1, a2,b2, construct an equation to determine if both of them has at least one ...
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43 views

Binomial distribution or probability intersection

I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = ...
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internal rectangle area intersected by a circle

I need to compute the internal rectangle area intersected by a circle like (the blue area) on these 3 examples: I know every vertex (x,y) coordinate and then their distance from circle center but ...
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1answer
57 views

Finding Unions and intersections given two probabilities.

I am currently trying to find the unions and probabilities given: A = .2 and B = .6. P($A\cap B$) = .12 And am looking to find ...
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2answers
84 views

Find normal to ellipse through arbitrary points

I want to find the normal to ellipse through an arbitrary point. There is an array of points located arround a given ellipse (but not on ellipse curve). What I want to find is the normal of each of ...
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1answer
43 views

Countable intersections in sigma-rings

A problem in Friedmans Analysis is to show that a $\sigma$-ring is closed under countable intersections. Let the $\mathcal{R}$ be the $\sigma$-ring. I tried to solve it as follows: Since $A - B \in ...