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How do you prove that a finite set of points cannot have any accumulation points in a real analysis?
Wiki User
∙ 10y ago
If you have a finite set of points (call them A1, A2, A3...), then you have a finite set of distances to the points.
So for any point B, simply pick a distance D that's smaller than the distance between B and A1, the distance between B and A2, and so on. (This is possible, since there a finite number of points.)
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Since there are no points within distance D of B (because this is how you chose D), point B can not be an accumulation point (because an accumulation point must have points within any given distance of it)
Wiki User
∙ 10y ago
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How do you prove that a finite set of points cannot have any accumulation points in a complex analysis?
Complex analysis is a metric space so neighborhoods can be described as open balls. Proof follows a. Assume that the set has an accumulation point call it P. b. An accumulation point is defined as a point in which every neighborhood (open ball) around P contains a point in the set other than P. c. Since P is an accumulation point, I can choose an open ball around P that has a diameter less than the minimum distance between P and all elements of the finite set. Therefore there exists a neighbor hood around P which contains only P. Therefore P is not an accumulation point.
What graph has a finite or limited number of data points is?
A discrete graph.
Is a ray line segment or line longer?
Let's think of a line segment as a finite set of points. Along those lines, (pun intended) think of a ray and a line as infinite sets of points. Then we think of longer in terms of the size of the set. So for example, a 2 inch line segment would be longer than a 1 inch line segment because we can have more points in the set which is made of the two inch segment. The ray and the line are the same size since they both can be viewed as sets containing an infinite number of points. The line segment being a finite set is smaller than the other two.
What is the center of the circle that has endpoints are 2 7 and -6 -1?
A circle cannot have end points!
What is 4.12.10 with the scientific notation expanded?
The expression in the question contains two decimal points and so cannot be a number.
How do you prove that a finite set of points cannot have any accumulation points in a complex analysis?
Complex analysis is a metric space so neighborhoods can be described as open balls. Proof follows a. Assume that the set has an accumulation point call it P. b. An accumulation point is defined as a point in which every neighborhood (open ball) around P contains a point in the set other than P. c. Since P is an accumulation point, I can choose an open ball around P that has a diameter less than the minimum distance between P and all elements of the finite set. Therefore there exists a neighbor hood around P which contains only P. Therefore P is not an accumulation point.
Is an adherent point an accumulation point?
No, not all adherent points are accumulation points. But all accumulation points are adherent points.
How could one perform finite element analysis?
Finite element analysis is used to evaluate materials to determine their performance under particular stresses, in order to determine weak points in a structure. There are FEA software programs available to record the calculations, but performing the analysis correctly may require training in mechanical engineering.
What do you call a set of numbers with an exact number of points?
This is called a discrete set (all points isolated) or a finite set. Finite sets are always discrete.
In calculus do open sets have accumulation points?
Yes, every point in an open set is an accumulation point.
Is the set of points in a line finite or infinite?
Infinite.
Is a geometric figure is a set of a finite number of points?
Yes
Is it tue that it two planes intersect their points of intersection are finite?
No.
Let f be a function with a finite domain The graph of f is necessarily made up of a finite number of points?
true
What graph has a finite or limited number of data points is?
A discrete graph.
Why shouldn't string stretch?
a line is a set of infinite points. a stretched string is finite.
In Calculus what is an accumulation point?
An accumulation point, or limit point, for a set S is a point x (not necessarily in S) such that any open set containing x also contains a point (distinct from x) that's in S. More intuitively, it means that by choosing points in S, we can get as close as we want to x without actually reaching it. For example, consider the set S={1,1/2,1/3,1/4,...} (in the real numbers). 0 is an accumulation point for S, because any open set containing 0 would have to contain all between 0 and some ε>0, which would include a point (actually, an infinite amount of points) in S. But 1/5, for example, is not an accumulation point for S, because we can take the open interval (11/60,9/40) which doesn't contain any points in S other than 1/5. Not all sets have an accumulation point. For example, any set of a finite amount of real numbers can't have an accumulation point. Another example of a set without an accumulation point is the integers (as a subset of the real numbers). However, over the real numbers, any bounded infinite set has an accumulation point. In a general topological space, any infinite subset of a compact set has an accumulation point.
