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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Max length of meaningful combinations of union and intersection of three sets.

Given three sets: $X, Y, Z$ and the two set operations: union and intersection. What is the maximum length of a 'formula' which is not reducible to a shorter formula. Eg. the formula $(X \cap Y) ...
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28 views

The intersections of two equations

I have these two functions, that I must solve. When I plot them, I see there are 4 intersections: $$(1,0),(0,1),(-1,0),(0,-1)$$ But how do you solve these??
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98 views

Vector Tangent to Curve of Intersection

I am having problems solving this. Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$. I'm able to do this kind of thing using ...
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2answers
339 views

Probability of limsup

Let $A_1, A_2, A_3, \dots$ be a sequence of independent events on $\left (\Omega, \mathbb A, \mathbb P\right )$ such that $\mathbb P(A_n) < 1$ and $\mathbb P\left ...
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0answers
56 views

Finding the equation of a plane

I need to find the equation for the plane which goes through the intersection line of the planes $x-z=1$ and $y+2z=3$, and perpendicular to the plane $x+y-2z=1$. What I got so far - to find the ...
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2k views

Finding the coordinates of a point of intersection from a pair of parametric equations.

A curve is given by: $$x = 2t + 3 $$ $$y = t^3 - 4t$$ The point $A$ has parameter $t = -1$. Line l is a tangent to the curve at $A$. Line l cuts the curve at point $B$. Find the value of $t$ at ...
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1answer
278 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
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0answers
68 views

Equation to calculate intersected cells in a grid, given a selection rectangle

I am looking for an equation which will calculate which cells (returned either as a pure index or as a row/col) are intersected by a selection rectangle when provided with the box coordinates of each ...
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1answer
72 views

Ideal gen. by a set S = Intersection over ideals containing S

I am trying to prove the following statement: Let R be a ring and $I=\{\sum_{i=1}^n a_i x_i : a_i\in R\}$ the ideal generated by $S=\{x_1,\ldots, x_n\}$. Then $I$ is the intersection of ideals $J$ in ...
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1answer
125 views

Find intersection of two ranges on line

I have two ranges on 2D line defined by edge points. How to get intersection of these ranges with minimum operations? p1-p2 - first range p3-p4 - second range I want to get p3-p2 in this case ...
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1answer
984 views

Proof of Cartesian product intersection

How to prove $( A \times B ) \cap ( C \times D ) = ( A \cap C ) \times ( B \cap D )$ ?
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How to find intersection with x/y axis

As my question says, how do I find intersection with x/y axis. For example, if given function is f(x)=x^3+x^2-x-1, how do I find the intersection with x and y axis. Right now, I only know that when ...
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2answers
2k views

How to find an intersection of a 2 vector subspace?

Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$. how to get thier intersection?
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4answers
134 views

Finding the area of region inside region

I have two general parallelograms each defined by four vertices (the corners) in $\mathbb R^2$. I want to find the intersecting area of them. How would I go about doing this? I've thought for awhile, ...
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2answers
230 views

Intersection and complement proof

I'm trying to prove that if $A \cap B = A \cap C$ then $A \cap \overline{B} = A \cap \overline{C}$. I've tried several manipulations, but I can't get to it.
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4answers
302 views

May a group have two disjoint subgroups?

Is there any group $(G,+)$, that has $2$ or more subgroups $(H,+),\; (I,+)$, where $$H \cap I = \emptyset?$$ I believe not, because both $H$ and $I$ must contain neutral element. Is my assumption ...
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1answer
58 views

How do I best weigh the commonality between sets weighted to the size of the sets

I have about 350 online petitions, each of which has between 250 and 25,000 signatures. For any two petitions, I can easily measure how many individual signatories have signed both of them. I want to ...
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2answers
441 views

Determine the coordinate of the point where line and circle collide/intersect - how to solve for x

I would like to determine the point $C$ in this image: (assume I have radius value). After the hours of research and refreshing some memories from school days, I've got: Please assume... Point: ...
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1answer
168 views

Does a generalized intersection test exist? If yes, how does it work?

I'm looking for an algorithm to test if an N-dimensional object (defined by the convex hull of N+1 vertices) and an M-dimensional object (defined by the convex hull of M+1 vertices) intersect within ...
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1answer
114 views

Less restrictive set intersection

I have a question that may be trivial but I just can't find an appropriate answer on the Internet. The inclusion-exclusion principle can be used to discern the cardinality of the union among sets ...
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1answer
112 views

Deducing that “the probability of the intersection is (or is not) the product of the probabilities” from knowledge about other intersections

Let $A_1, A_2, \ldots, A_n$ be a collection of events in a probability space. There are $2^n - n - 1$ subsets S of $\{1, 2, \ldots, n\}$ for which we may or may not have $P(\bigcap_{j \in S}A_j) = ...
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72 views

Looking for a proper proof that two sets with sub sets are equivalent

Basically I would just like to know how to prove the following equation $$ A \subseteq B \cup C \iff A \cap \overline B \subseteq C $$ I understand that I have to prove that the left-hand side ...
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1answer
32 views

Intersection and derivatives

If two functions $f$ and $g$ are such that: $f(0) = g(0)$ $\forall x > 0,\ f^\prime(x) > g^\prime(x)$ is it true that $f\neq g$ for any $x>0$? I believe so, but I don't know how to ...
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3answers
658 views

Basis for intersection of subspaces

In $\mathbb{R}^4$ find a basis of $L_1\cap L_2$ where $$L_1 = \operatorname{span} \{ (1,2,0,3),(1,1,-1,2),(0,1,1,1) \}$$ $$L_2 = \operatorname{span} \{ (2,0,-2,1), (3,1,0,2),(4,2,2,3) \}$$ I tried ...
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2answers
104 views

How do I calculate a point on each of three circles that have specific distance to each other?

I am trying to write code for a computer simulator. I need to simulate a complex mechanism where each link has a known length and the ends of the links are connected to a triangle. I would like help ...
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1answer
233 views

Intersection of plane with quadratic regression surface

A quadratic OLS regression with two predictors is defined as: $Z = b_0 + b_1X + b_2Y + b_3X^2 + b_4XY + b_5Y^2 + e$ (1) If this regression surface is plotted, it can look like this ($b_0=10, ...
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1answer
50 views

How many faces can have at most the intersection of two rectangular frustums?

In a 3D context, I want to evaluate the intersection of two rectangular frustums. The intersection of those two frustums will be a convex polytope, I think. What will be the maximum number of faces ...
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1answer
291 views

Non Self Intersecting Polygons?

Given a set of n points is it always possible to construct a non self intersecting polygon?
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1answer
365 views

Algorithm to find the basis of intersection of subspaces without gaussian elimination.

Is there an algorithm to find the basis of intersection of subspaces $A_1$ and $A_2$, if we have the bases of subspaces $A_1$ and $A_2$, without using Gaussian elimination? Thanks.
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313 views

Can you help me solve these questions related to a Logical theory?

In a group of 200 people, number of people having at least primary education (assuming - Category I): number of people having at least middle school education (Category II): number of people having ...
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1answer
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Intersection between a rectangle and a circle?

I have a poor working knowledge of math. I would like to calculate collision detection between a 2D circle and a 2D rectangle for a simple game of Pong. I thought of splitting the 2D rectangle into 4 ...
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3answers
444 views

Equation to determine radius for a circle that should intersect a given point?

Simple question. I tried Google but I don't know what search keywords to use. I have two points on a 2d plane. Point 1 = x1 and y1, and Point 2 = x2 and y2. I'd like to draw a circle around Point ...
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0answers
423 views

regular expression and intersection

I have this language L that contains only one string: $a_{1}a_{1}a_{2}a_{1}a_{1}a_{2}a_{3}a_{1}a_{1}a_{2}a_{1}a_{1}a_{2}a_{3} ....a_{n}...a_{n}$ written more concisely ...
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4answers
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How can I find the points at which two circles intersect?

Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
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1answer
284 views

Intersection of a line segment and a paraboloid in 3D

Suppose I have a line segment $L$ in 3D: $$x=a_1(1-t)+b_1t$$ $$y=a_2(1-t)+b_2t$$ $$z=(a_1^2+a_2^2-k_1^2)(1-t)+(b_1^2+b_2^2-k_2^2)t$$ Because $L$ is line segment then $0\leq t\leq 1$. And defining ...
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1answer
581 views

Automatic calculation of the intersection of discrete curves

first of all, let me apologize for a poor math-english translation, I'll try my very best. I have the following situation: I have over 16.000 data files which I generated from a biometric ...
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621 views

Ray VS Line intersection

Continue vector vs plane intersection Image How to determine Ray (one point R(rx;ry) and alpha with OX) with line (two points A(ax;ay) and B(bx;by)) intersection?
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1answer
87 views

Vector VS Plane intersection

Could You help me with task: From point $M(3,5)$ that belongs to plane: $A(0,0), B(0,10), C(20,10), D(20,0)$, comes out vector $V$ at an angle a(with $OX$). Need to find point $X(x,y)$ at which he ...
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2answers
149 views

“Good” closure conditions

[Attention! This question requires some reading and it's answer probably is in form of a "soft-answer", i.e. it can't be translated into a hard mathematical proposition. (I hope I haven't scared away ...
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3answers
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How do I calculate the intersection(s) of a straight line and a circle?

The basic equation for a straight line is $y = mx + b$, where $b$ is the height of the line at $x = 0$ and $m$ is the gradient. The basic equation for a circle is $(x - c)^2 + (y - d)^2 = r^2$, where ...
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2answers
116 views

Angular radius of a sphere

Given a sphere with radius $r$ about a point $c$, what's the apparent angular radius $\alpha$ of that sphere from point $P$? In other words, if $\vec{o} = c - P$, what's the maximum angle another ...
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1answer
228 views

Volume of parallelepiped intersection with oblique plane

Is there an algorithm or program capable to find the volume resulting when a parallelepiped is intersected by an oblique plane? What is most important, although, it's not specified in title is: given ...
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Give an example of open, nested sets such that the intersection is closed nonempty.

Give an example of open, nested sets such that the intersection is closed nonempty. I will ask questions if I am in doubt of the example provided. Thank you!
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1answer
197 views

Union, intersection and complement of a set

Is this equation true? $$\mathcal C \bigcap_{M\in A}M=\bigcup_{M\in A}\mathcal CM$$ where C is the complement, M is a set, A is a set of sets. I don't know how to start proving or disproving it ...
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Divide circle into 9 pieces of equal area

I'd like to divide a unit circle disk into nine parts of equal area, using circle arcs as delimiting lines. The whole setup should be symmetric under the symmetry group of the square, i.e. 4 mirror ...
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1answer
1k views

Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx

Let L be the line given by the parametric equations x = 1 + 2t, y = −1 − t, z = 3t Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx
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1answer
597 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
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1answer
137 views

Build equation of a curve with set of coordinates

I need to calculate the intersection of two curves. I do not have the equation of the curves, but I will have a finite set of coordinates. Is there a way to build the equation for this curve based ...
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2answers
194 views

Bilinear Interp Surface Intersection

Edit: I rephrased the question to make it clearer, sorry! I'm trying to solve for the intersection of two surfaces in three spatial dimensions and time. Consider each of these surfaces as some ...
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130 views

Intersection of a hyperplane and a quadric surface.

I have $r, d \in \mathbb{R}^n$. I want to find all $y \in \mathbb{R}^n$ such that $r \cdot y = 0$ and $d \cdot \langle y_1^2, \dots, y_n^2 \rangle = 0$. The trivial $y = 0$ solution is not terribly ...