若集合M={y|y=x2，x∈Z}，N={x|x2-6x-27≥0，x∈R}，全集U=R，则M∩（?UN）的真子集的个数_答案_百度高考
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若集合M={y|y=x2，x∈Z}，N={x|x2-6x-27≥0，x∈R}，全集U=R，则M∩（?UN）的真子集的个数是（　　）
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下载知乎 App设全集U＝R，M＝{x|y＝x2?4}，N＝{x|2x?1≥1}都是U的子集（如图所示），则阴影部分所示的集合是_______百度知道
设全集U＝R，M＝{x|y＝x2?4}，N＝{x|2x?1≥1}都是U的子集（如图所示），则阴影部分所示的集合是______
4}; border-font-size: hidden.hiphotos设全集U＝R?1≥1}都是U的子集（如图所示）: background-clip，M＝{x|y＝2x://b，则阴影部分所示的集合是______．
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等待您来回答From Wikipedia, the free encyclopedia
In , the Hausdorff distance, or Hausdorff metric, also called –Hausdorff distance, measures how far two
are from each other. It turns the set of
subsets of a metric space into a metric space in its own right. It is named after .
Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set.
It seems that this distance was first introduced by Hausdorff in his book , first published in 1914.
Components of the calculation of the Hausdorff distance between the green line X and the blue line Y.
Let X and Y be two non-empty subsets of a metric space (M, d). We define their Hausdorff distance d H(X, Y) by
{\displaystyle d_{\mathrm {H} }(X,Y)=\max\{\,\sup _{x\in X}\inf _{y\in Y}d(x,y),\,\sup _{y\in Y}\inf _{x\in X}d(x,y)\,\}{\mbox{,}}\!}
where sup represents the
and inf the .
Equivalently
{\displaystyle d_{H}(X,Y)=\inf\{\epsilon \geq 0\,;\ X\subseteq Y_{\epsilon }\ {\mbox{and}}\ Y\subseteq X_{\epsilon }\}}
{\displaystyle X_{\epsilon }:=\bigcup _{x\in X}\{z\in M\,;\ d(z,x)\leq \epsilon \}}
that is, the set of all points within
{\displaystyle \epsilon }
of the set
{\displaystyle X}
(sometimes called the
{\displaystyle \epsilon }
-fattening of
{\displaystyle X}
or a generalized ball of radius
{\displaystyle \epsilon }
{\displaystyle X}
It is not true in general that if
{\displaystyle d_{H}(X,Y)=\epsilon }
{\displaystyle X\subseteq Y_{\epsilon }\ {\mbox{and}}\ Y\subseteq X_{\epsilon }}
For instance, consider the metric space of the real numbers
{\displaystyle \mathbb {R} }
with the usual metric
{\displaystyle d}
induced by the absolute value,
{\displaystyle d(x,y):=|y-x|,\quad x,y\in \mathbb {R} }
{\displaystyle X:=(0,1]\quad {\mbox{and}}\quad Y:=[-1,0)}
{\displaystyle d_{H}(X,Y)=1\ }
{\displaystyle X\nsubseteq Y_{1}}
{\displaystyle Y_{1}=[-2,1)\ }
{\displaystyle 1\in X}
In general, dH(X,Y) may be infinite. If both X and Y are , then dH(X,Y) is guaranteed to be finite.
dH(X,Y) = 0 if and only if X and Y have the same .
For every point x of M and any non-empty sets Y, Z of M: d(x,Y) ≤ d(x,Z) + dH(Y,Z), where d(x,Y) is the distance between the point x and the closest point in the set Y.
|diameter(Y)-diameter(X)| ≤ 2 dH(X,Y).
If the intersection X∩Y has a non-empty interior, then there exists a constant r&0, such that every set X′ whose Hausdorff distance from X is less than r also intersects Y.
On the set of all non-empty subsets of M, dH yields an extended .
On the set F(M) of all non-empty compact subsets of M, dH is a metric.
If M is , then so is F(M).
() If M is compact, then so is F(M).
of F(M) depends only on the topology of M, not on the metric d.
The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function d(x, y) in the underlying metric space M, as follows:
Define a distance function between any point x of M and any non-empty set Y of M by:
{\displaystyle d(x,Y)=\inf\{d(x,y)|y\in Y\}\ }
For example, d(1, [3,6]) = 2 and d(7, [3,6]) = 1.
Define a distance function between any two non-empty sets X and Y of M by:
{\displaystyle d(X,Y)=\sup\{d(x,Y)|x\in X\}\ }
For example, d([1,7], [3,6]) = sup{d(1, [3,6]), d(7,[3,6])} = sup{d(1,3), d(7,6)} = 2.
If X and Y are compact then d(X,Y) d(X,X)=0; and d inherits the
property from the distance function in M. As it stands, d(X,Y) is not a metric because d(X,Y) is not always symmetric, and d(X,Y) = 0 does not imply that X = Y (It does imply that
{\displaystyle X\subseteq Y}
). For example, d([1,3,6,7], [3,6]) = 2, but d([3,6], [1,3,6,7]) = 0. However, we can create a metric by defining the Hausdorff distance to be:
{\displaystyle d_{\mathrm {H} }(X,Y)=\max\{d(X,Y),d(Y,X)\}\,.}
In , the Hausdorff distance can be used to find a given template in an arbitrary target image. The template and image are often pre-processed via an
giving a . Next, each 1 (activated) point in the binary image of the template is treated as a point in a set, the "shape" of the template. Similarly, an area of the binary target image is treated as a set of points. The algorithm then tries to minimize the Hausdorff distance between the template and some area of the target image. The area in the target image with the minimal Hausdorff distance to the template, can be considered the best candidate for locating the template in the target. In
the Hausdorff distance is used to measure the difference between two different representations of the same 3D object particularly when generating
for efficient display of complex 3D models.
A measure for the dissimilarity of two
is given by Hausdorff distance up to isometry, denoted DH. Namely, let X and Y be two compact figures in a metric space M (usually a ); then DH(X,Y) is the infimum of dH(I(X),Y) along all
I of the metric space M to itself. This distance measures how far the shapes X and Y are from being isometric.
is a related idea: we measure the distance of two metric spaces M and N by taking the infimum of dH(I(M),J(N)) along all isometric embeddings I:M→L and J:N→L into some common metric space L.
(2005). Variational Analysis. Springer-Verlag. p. 117.  .
Munkres, James (1999).
(2nd ed.). Prentice Hall. pp. 280–281.  .
Henrikson, Jeff (1999).
(PDF). MIT Undergraduate Journal of Mathematics: 69–80. Archived from
(PDF) on June 23, 2002.
(1993). Fractals Everywhere. Morgan Kaufmann. pp. Ch. II.6.  .
Cignoni, P.; Rocchini, C.; Scopigno, R. (1998). "Metro: Measuring Error on Simplified Surfaces". Computer Graphics Forum. 17 (2): 167–174.
A short tutorial on how to compute and visualize the Hausdorff distance between two triangulated 3D surfaces using the open source tool .
MATLAB code for Hausdorff distance:}