# Tagged Questions

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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### Covering with most possible equal size subsets having pairwise singleton intersections

Assume I have a non-empty finite set $S$ with $x=|S|$. I want to divide the set $S$ into subsets $S_1, S_2, .., S_n$ (Edit: Yes, $S = \cup S_i$, and I'm embarrassed that I forgot to include that) such ...
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### Determinant in Line-Line Intersection

Assume we have two equations of a line, $A_1 x + B_1y = C_1$ and $A_2 x + B_2y = C_2$ Now we multiply the first equation by $B_2$ and the second by $B_1$ to obtain (1) $A_1B_2x + B_1B_2y = C_1B_2$ ...
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### Line Triangle Intersection Mathematics

I am following the math in the book Real Time Collision Detection by Christer Ericson. On pages 184 thru 188, he discusses how to test for an intersecting line against a triangle. I replicated the ...
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### Probability for intersection of sets

(It is an easier - hopefully - formulation of a question I asked previously). Assume that we are given $J$ sets, each set with $n_j$ elements ($J$ and $n_j$ for each $j \le J$ are known). The ...
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### Is there an equation to find the intersection of 3 circles without complex steps?

Is there a way to find the intersection 3 circles without substituting and solving the equations into each other? The reason is because I am making a trilateration program, so I won't really be able ...
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### Prove $S\cap S ^\bot=\{0\}$

Let $S$ be a nonempty subset of the inner product space $V$ and $S^\bot$ be the orthogonal complement of $S$. Prove $S\cap S ^\bot=\{0\}$
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### How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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### How to express exclusive intersections?

I have a function $f:\mathbb{R}^2 \rightarrow\mathbb{R}, \ \mathbf{x} \mapsto \sum_{i = 1}^{n} w_i \mathbf{1}_{A_i}(\mathbf{x})$ for $w_i \in \mathbb{R}$ and the $A_i$s are allowed to intersect. Thus ...
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### Probability for intersection of size $i$ between $k$ sets of size up to $N$

Assume that the elements in each set are sampled uniformly from $r$ possible elements ($N \le r$). It is also given that an element does not appear twice in a set.
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### 3D Circle/ground intersection

This one stumps me: A circle in 3D space given by its center = $(0.15, 0.5, 1.0)$, its radius $=64$ and an orientation vector that points away from the circle's plane $(0.251, -0.796, 0.551)$ How ...
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### Intersection of intervals

Let $a\in\mathbb{N}_{\geq3}$. How can one prove that $$\bigcap_{i = 1}^{a} \bigcup_{j = 0}^{i-1} \left[\frac{1+aj}{i},\frac{a(j+1)-1}{i}\right] = \varnothing,$$ where $\varnothing$ is the empty set? ...
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### Number of subsets the cardinality of whose intersections with some other subsets are known

$A$ is a non-empty finite set. $A_1,A_2,\ldots,A_n$ are subsets of $A$. How many subsets $B$'s of $A$ are there that satisfy that $|B\cap A_i|=a_i,\forall 1\leq i\leq n$, where $a_i\geq 0$'s are given ...
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### Ray-Lens Intersection

So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector. I also have a spherical lens with aperture ...
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### Ray Disk intersection

So if I have a ray parameterized as $O + tD$ where $O$ is the origin, $D$ is the direction and $t$ is the parameter variable and a flat circular disk with a center point $P$ in 3D space and a radius ...
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### Intersection between sphere and cylinder

I have a sphere and a cylinder. I have the center and the radius of each of them. the sphere: radius = $r_1$ center = $(x_1,y_1,z_1)$ the cylinder: radius = $r_2$ height = $h_2$ center = ...
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### infinite intersection of sets

I'm trying to proof this identity: $[a,b]\equiv \bigcap_{n=1}^\infty [a,b+\frac{1}{n}) \equiv \bigcap_{n=1}^\infty (a-\frac{1}{n} , b]$ I already try to use De-Morgan's lows but with no success . ...
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### Probability that subsets intersect

Given a set $N$ I would like to calculate the probability that two arbitrarily chosen and equally likely subsets $K\subseteq N$ and $J\subseteq N$ both of fixed size intersect. Let's say $n=\#N$, ...
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### proof that intersection of two conic sections will intersect at at least two points.

In the following equation $\rho(x,y)$ returns a constant value for a given coordinate. $\mathbf n$ is the normal vector to the surface of the form $[P,Q,-1]$ and $s$ is a direction vector. ...
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### Identity Law - Set Theory

I'm trying to wrap my head around the Identity Law, but I'm having some trouble. My lecture slides say: $$A \cup \varnothing = A$$ I can understand this one. $A$ union nothing is still $A$. In the ...
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### 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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### 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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### $2D$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...