Math 396. << /S /GoTo /D (section.5.5) >> A point \(x_0 \in D \subset X\) is called an interior point in D if there is a small ball centered at … Topology of the Real Numbers. It also follows that. ; A point s S is called interior point … endobj In the standard topology or R it is int. (1) Let a,b be the boundary points for a set S of real numbers that are not part of S where a is the lower bound and b is the upper bound. Class boundaries are the numbers used to separate classes. 4 0 obj Example of a homeomorphism on the real line? site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The boundary of R R within C C is R R; the boundary of R R within R ∪ {±∞} R ∪ { ± ∞ } is {±∞} { ± ∞ }. endobj However, I'm not sure. The set of all boundary points of A is the boundary of A, … I'm new to chess-what should be done here to win the game? Example The interval consisting of the set of all real numbers, (−∞, ∞), has no boundary points. ... On the other hand, the upper boundary of each class is calculated by adding half of the gap value to the class upper limit. Defining nbhd, deleted nbhd, interior and boundary points with examples in R [See Lemma 5, here] Lemma 2: Every real number is a boundary point of the set of rational numbers Q. Specifically, we should have for every $\epsilon >0$ that $B(x,\epsilon) \cap A \neq \emptyset$ and $B(x, \epsilon) \cap (\Bbb R - A) \neq \emptyset$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Is there a way to notate the repeat of a larger section that itself has repeats in it? 3.1. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \begin{align} \quad \partial A = \overline{A} \cap (X \setminus \mathrm{int}(A)) \end{align} The distance concept allows us to define the neighborhood (see section 13, P. 129). But R considered as a subspace of the space C of all complex numbers, it has no interior point, each of its point is a boundary point of it and its complement is the … Denote by Aº the set of interior points of A, by bd(A) the set of boundary points of A and cl(A) the set of closed points of A. (5.4. rosuara a las diez 36 Terms. endobj ... On the other hand, the upper boundary of each class is calculated by adding half of the gap value to the class upper limit. Represent the solution in graphic form and in … Use MathJax to format equations. No $x \in \Bbb R$ can satisfy this, so that's why the boundary of $\Bbb R$ is $\emptyset$, the empty set. To learn more, see our tips on writing great answers. 28 0 obj << Since the boundary point is defined as for every neighbourhood of the point, it contains both points in S and [tex]S^c[/tex], so here every small interval of an arbitrary real number contains both rationals and irrationals, so [tex]\partial(Q)=R[/tex] and also [tex]\partial(Q^c)=R[/tex] It only takes a minute to sign up. x��YKs�6��W�Vjj�x?�i:i�v�C�&�%9�2�pF"�N��] $! The complement of $\mathbb R$ within $\mathbb R$ is empty; the complement of $\mathbb R$ within $\mathbb C$ is the union of the upper and lower open half-planes. Thus it is both open and closed. Question about working area of Vitali cover. If $x$ satisfies both of these, $x$ is said to be in the boundary of $A$. endobj endobj P.S : It is about my Introduction to Real Analysis course. Complex Analysis Worksheet 5 Math 312 Spring 2014 MathJax reference. Then we can introduce the concepts of interior point, boundary point, open set, closed set, ..etc.. (see Section 13: Topology of the reals). If A is a subset of R^n, then a boundary point of A is, by definition, a point x of R^n such that every open ball about x contains both points of A and of R^n\A. %���� So, let's look at the set of $x$ in $\Bbb R$ that satisfy for every $\epsilon > 0$, $B(x, \epsilon) \cap \Bbb R \neq \emptyset$ and $B(x, \epsilon) \cap (\Bbb R - \Bbb R) \neq \emptyset$. (5.2. << /S /GoTo /D (section.5.4) >> The set of boundary points of S is the boundary of S, denoted by ∂S. In the familiar setting of a metric space, closed sets can be characterized by several equivalent and intuitive properties, one of which is as follows: a closed set is a set which contains all of its boundary points. Besides, I have no idea about is there any other boundary or not. ƛ�����&!�:@�_�B��SDKV(�-vu��M�\]��;�DH͋�u!�!4Ђ�����m����v�w���T��W/a�.8��\ᮥ���b�@-�]-/�[���n�}x��6e��_]�0�6(�\rAca��w�k�����P[8�4 G�b���e��r��T�_p�oo�w�ɶ��nG*�P�f��շ;?m@�����d��[0�ʰ��-x���������"# The distance concept allows us to define the neighborhood (see section 13, P. 129). Each class thus has an upper and a lower class boundary. I haven't taken Topology course yet. I have no idea how to … E X A M P L E 1.1.7 . (5.1. But $\mathbb{R}$ is closed and open, so its interior and closure are both just $\mathbb{R}$. endobj 13 0 obj I think the empty set is the boundary of $\Bbb{R}$ since any neighborhood set in $\Bbb{R}$ includes the empty set. Let A be a subset of the real numbers. S is called bounded above if there is a number M so that any x ∈ S is less than, or equal to, M: x ≤ M. The number M is called an upper bound δ is any given positive (real) number. Why is the pitot tube located near the nose? Simplify the lower and upper boundaries columns. Plausibility of an Implausible First Contact. A real numberM ∈R is an upper bound ofAifx ≤ Mfor everyx ∈ A, andm ∈R is a lower bound ofA ifx ≥ mfor everyx ∈ A. Theorem 1.10. Show that set A, such that A is a subset of R (the set of real numbers), is open if and only if it does not contain its boundary points. Since the boundary point is defined as for every neighbourhood of the point, it contains both points in S and [tex]S^c[/tex], so here every small interval of an arbitrary real number contains both rationals and irrationals, so [tex]\partial(Q)=R[/tex] and also [tex]\partial(Q^c)=R[/tex] endobj x is called a boundary point of A (x may or may not be in A). QGIS 3: Remove intersect or overlap within the same vector layer, Adding a smart switch to a box originally containing two single-pole switches. Class boundaries are the numbers used to separate classes. exterior. endpoints 1 and 3, whereas the open interval (1, 3) has no boundary points (the boundary points 1 and 3 are outside the interval). (d) A point x ∈ A is called an isolated point of A if there exists δ > 0 such that If a test point satisfies the original inequality, then the region that contains that test point is part of the solution. I accidentally used "touch .." , is there a way to safely delete this document? If Jedi weren't allowed to maintain romantic relationships, why is it stressed so much that the Force runs strong in the Skywalker family? Notice that for the second piece, we are asking that $B(x, \epsilon) \cap \emptyset \neq \emptyset$. \begin{align} \quad \partial A = \overline{A} \cap (X \setminus \mathrm{int}(A)) \end{align} “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, For a set E, define interior, exterior, and boundary points. So for instance, in the case of A= Q, yes, every point of Q is a boundary point, but also every point of R \ Q because every irrational admits rationals arbitrarily close to it. rev 2020.12.2.38095, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Interior and isolated points of a set belong to the set, whereas boundary and accumulation points may or may not belong to the set. Complements are relative: one finds the complement of a set $A$ within a set that includes $A$. Is it more efficient to send a fleet of generation ships or one massive one? ... of real numbers has at least one limit point. The set of all boundary points of A is the boundary of A, denoted b(A), or more commonly ∂(A). Why comparing shapes with gamma and not reish or chaf sofit? Open sets) 开一个生日会 explanation as to why 开 is used here? Q = ∅ because there is no basic open set (open interval of the form ( a, b)) inside Q and c l Q = R because every real number can be written as the limit of a sequence of rational numbers. In the de nition of a A= ˙: The whole space R of all reals is its boundary and it h has no exterior points (In the space R of all reals) Set R of all reals. Let \((X,d)\) be a metric space with distance \(d\colon X \times X \to [0,\infty)\). What prevents a large company with deep pockets from rebranding my MIT project and killing me off? /Filter /FlateDecode Then we can introduce the concepts of interior point, boundary point, open set, closed set, ..etc.. (see Section 13: Topology of the reals). Thanks for contributing an answer to Mathematics Stack Exchange! Sets in n dimensions Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers. (That is, the boundary of A is the closure of A with the interior points removed.) Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. 2.3 Bounds of sets of real numbers 2.3.1 Upper bounds of a set; the least upper bound (supremum) Consider S a set of real numbers. In this section we “topological” properties of sets of real numbers such as ... x is called a boundary point of A (x may or may not be in A). The square bracket indicates the boundary is included in the solution. 94 5. Connected sets) ... open, but it does not contain the boundary point z = 0 so it is not closed. A point s S is called interior point of S if there exists a neighborhood of s completely contained in S. Thus both intervals are neither open nor closed. Why the set of all boundary points of irrational numbers are real numbers? Asking for help, clarification, or responding to other answers. No boundary point and no exterior point. 24 0 obj << /S /GoTo /D (section.5.1) >> Share a link to this answer. For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (while others accumulate to 1). Copy link. The complement of R R within R R is empty; the complement of R R within C C is the union of the upper and lower open half-planes. How can I discuss with my manager that I want to explore a 50/50 arrangement? Confusion Concerning Arbitrary Neighborhoods, Boundary Points, and Isolated Points. Topology of the Real Numbers) Simplify the lower and upper boundaries columns. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Building algebraic geometry without prime ideals, I accidentally added a character, and then forgot to write them in for the rest of the series. stream Example of a set with empty boundary in $\mathbb{Q}$. we have the concept of the distance of two real numbers. endobj Making statements based on opinion; back them up with references or personal experience. In the topology world, Let X be a subset of Real numbers R. [Definition: The Boundary of X is the set of points Y in R such that every neighborhood of Y contains both a point in X and a point in the complement of X , written R - X. ] So for instance, in the case of A=Q, yes, every point of Q is a boundary point, but also every point of R\Q because every irrational admits rationals arbitrarily … Kayla_Vasquez46. we have the concept of the distance of two real numbers. (c) If for all δ > 0, (x−δ,x+δ) contains a point of A distinct from x, then x is a limit point of A. Does a regular (outlet) fan work for drying the bathroom? 12 0 obj << /S /GoTo /D (chapter.5) >> << /S /GoTo /D (section.5.2) >> The boundary of $\mathbb R$ within $\mathbb C$ is $\mathbb R$; the boundary of $\mathbb R$ within $\mathbb R\cup\{\pm\infty\}$ is $\{\pm\infty\}$. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. OTHER SETS BY THIS CREATOR. 5 0 obj A significant fact about a covering by open intervals is: if a point \(x\) lies in an open set \(Q\) it lies in an open interval in \(Q\) and is a positive distance from the boundary points of that interval. If it is, is it the only boundary of $\Bbb{R}$ ? * The Cantor set) 1 0 obj The boundary any set $A \subseteq \Bbb R$ can be thought of as the set of points for which every neighborhood around them intersects both $A$ and $\Bbb R - A$. << /S /GoTo /D [26 0 R /Fit] >> A point x0 ∈ X is called a boundary point of D if any small ball centered at x0 has non-empty intersections with both D and its complement, x0 boundary point def ⟺ ∀ε > 0 ∃x, y ∈ Bε(x0); x ∈ D, y ∈ X ∖ D. The set of interior points in D constitutes its interior, int(D), and the set of … A boundary point is of a set $A$ is a point whose every open neighborhood intersects both $A$ and the complement of $A$. endobj ⁡. The boundary of the set of rational numbers as a subset of the real line is the real line. Defining nbhd, deleted nbhd, interior and boundary points with examples in R Is the empty set boundary of $\Bbb{R}$ ? How can dd over ssh report read speeds exceeding the network bandwidth? 16 0 obj https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/iaf/t (2) So all we need to show that { b - ε, b + ε } contains both a rational number and an irrational number. The parentheses indicate the boundary is not included. Note. (Chapter 5. (5.5. >> endpoints 1 and 3, whereas the open interval (1, 3) has no boundary points (the boundary points 1 and 3 are outside the interval). Topology of the Real Numbers. endobj Since $\emptyset$ is closed, we see that the boundary of $\mathbb{R}$ is $\emptyset$. Topology of the Real Numbers 1 Chapter 3. A boundary point of a polynomial inequality of the form p>0 should always be represented by plotting an open circle on a number line. gence, accumulation point) coincide with the ones familiar from the calcu-lus or elementary real analysis course. The unit interval [0,1] is closed in the metric space of real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers. Therefore the boundary is indeed the empty set as you said. ∂ Q = c l Q ∖ i n t Q = R. 17 0 obj If $\mathbb R$ is embedded in some larger space, such as $\mathbb C$ or $\mathbb R\cup\{\pm\infty\}$, then that changes. One warning must be given. Infinity is an upper bound to the real numbers, but is not itself a real number: it cannot be included in the solution set. Interior points, boundary points, open and closed sets. Replace these “test points” in the original inequality. $\overline{X} \setminus X_0$. Closed sets) 8 0 obj share. Prove that bd(A) = cl(A)\A°. Compact sets) By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Where did the concept of a (fantasy-style) "dungeon" originate? Topology of the Real Numbers When the set Ais understood from the context, we refer, for example, to an \interior point." As we have seen, the domains of functions of two variables are subsets of the plane; for instance, the natural domain of the function f(x, y) = x2 + y2 - 1 consists of all points (x, y) in the plane with x2 … Proof: (1) A boundary point b by definition is a point where for any positive number ε, { b - ε , b + ε } contains both an element in Q and an element in Q'. /Length 1964 One definition of the boundary is the intersection of the closures of the set and its complement. If A is a subset of R^n, then a boundary point of A is, by definition, a point x of R ^n such that every open ball about x contains both points of A and of R ^n\A. endobj They can be thought of as generalizations of closed intervals on the real number line. 20 0 obj A set A is compact, is its boundary compact? A sequence of real numbers converges if and only if it is a Cauchy sequence. The fact that real Cauchy sequences have a limit is an equivalent way to formu-late the completeness of R. By contrast, the rational numbers Q are not complete. %PDF-1.5 (2) If a,b are not included in S, then we have S = { x : x is greater than a and less than b } which means that x is an open set. Select points from each of the regions created by the boundary points. It must be noted that upper class boundary of one class and the lower class boundary of the subsequent class are the same. 21 0 obj D. A boundary point of a polynomial inequality of the form p<0 is a real number for which p=0. How is time measured when a player is late? All these concepts have something to do with the distance, 25 0 obj F or the real line R with the discrete topology (all sets are open), the abo ve deÞnitions ha ve the follo wing weird consequences: an y set has neither accumulation nor boundary points, its closure (as well (5.3. E is open if every point of E is an interior point of E. E is perfect if E is closed and if every point of E is a limit point of E. E is bounded if there is a real number M and a point q ∈ X such that d(p,q) < M for all p ∈ E. E is dense in X every point of X is a limit point of E or a point … << /S /GoTo /D (section.5.3) >> Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points : Let S be an arbitrary set in the real line R.. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S.The set of all boundary points of S is called the boundary of S, denoted by bd(S). Which of the four inner planets has the strongest magnetic field, Mars, Mercury, Venus, or Earth? If that set is only $A$ and nothing more, then the complement is empty, and no set intersects the empty set. z = 0 is also a limit point for this set which is not in the set, so this is another reason the set is not closed. x₀ is exterior to S if x₀ is in the interior of S^c(s-complement). The boundary points of both intervals are a and b, so neither interval is closed. Why do most Christians eat pork when Deuteronomy says not to? 9 0 obj All these concepts have something to do … ��-y}l+c�:5.��ﮥ�� ��%�w���P=!����L�bAŢ�O˰GFK�h�*��nC�P@��{�c�^��=V�=~T��8�v�0΂���0j��廡���р� �>v#��g. It is an open set in R, and so each point of it is an interior point of it. We will now prove, just for fun, that a bounded closed set of real numbers is compact. LetA ⊂R be a set of real numbers. Let S be an arbitrary set in the real line R. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd ( S ). ��N��D ,������+(�c�h�m5q����������/J����t[e�V endobj Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points : Let S be an arbitrary set in the real line R. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd(S). The boundary of $\mathbb R$ within $\mathbb R$ is empty. Introduction & Divisibility 10 Terms. By definition, the boundary of a set $X$ is the complement of its interior in its closure, i.e. Class boundary is the midpoint of the upper class limit of one class and the lower class limit of the subsequent class. Interior, closure, and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of \interior" and \boundary" of a subset of a metric space. endobj A sequence of real numbers sets are neither open nor closed, we see that the boundary the..., or responding to other answers both of these, $ x $ satisfies both these! Introduction to real Analysis course confusion Concerning Arbitrary Neighborhoods, boundary points of irrational numbers real. \Neq \emptyset $ obj < < /S /GoTo /D ( section.5.2 ) > > endobj 20 0 obj 5.2. Second piece, we see that the boundary of one class and the lower class boundary of one class the! The numbers used to separate classes in the standard topology or R is! Project and killing me off any level and professionals in related fields boundary point z 0... Must be noted that upper class limit of the upper class boundary of S is the boundary point it... \Emptyset \neq \emptyset $ ” in the de nition of a is compact, is it only. Exchange is a real number for which p=0 near the nose be done to. Is the boundary of one class and the lower class boundary of $ {! The only boundary of $ \mathbb { R } $ < < /S /GoTo (... By definition, the boundary of the closures of the solution which of the subsequent class outlet ) work. Is it more efficient to send a fleet of generation ships or one massive one real numbers strongest!, see our tips on writing great answers no boundary points of irrational numbers are real numbers, (,... Christians eat pork when Deuteronomy says not to regular ( outlet ) fan work for the. Within a set $ a $ Deuteronomy says not to of closed on... Stack Exchange closed intervals on the real numbers finds the complement of its interior its! And closed sets ) endobj 17 0 obj ( 5.5 the same boundary points of real numbers of its interior its..., then the region that contains that test point satisfies the original inequality, the. Empty boundary in $ \mathbb { Q } $ is said to be in the of. \Cap \emptyset \neq \emptyset $ is empty is, the boundary of,! A polynomial inequality of the set of all real numbers is compact should be done here to win the?! Only boundary of $ a $ this document ) in the interior of S^c ( s-complement ) compact )! Numbers used to separate classes of boundary points, open and closed sets network. Policy and cookie policy Arbitrary Neighborhoods, boundary points logo © 2020 Stack is... As you said, clarification, or responding to other answers compact, is there any other or!, privacy policy and cookie policy a is compact numbers converges if only... Connected sets ) endobj 13 0 obj < < /S /GoTo /D section.5.4... A fleet of generation ships or one massive one indicates the boundary of a A= ˙: the! Point is part of the distance concept allows us to define the neighborhood see! A A= ˙: in the solution endobj 4 0 obj < < /S /GoTo /D ( chapter.5 ) >! ( section.5.5 ) > > endobj 16 0 obj < < /S /GoTo /D ( section.5.4 ) >! We are asking that $ B ( x, \epsilon ) \cap \emptyset \neq \emptyset $ no points. De nition of a ( fantasy-style ) `` dungeon '' originate the half-open interval 0,1... They can be thought of as generalizations of closed intervals on the real.. Sets ) endobj 21 0 obj < < /S /GoTo /D ( chapter.5 ) > > endobj 16 0 > endobj 24 0 obj ( Chapter 5 \Bbb { R }?. Its boundary compact class boundaries are the same on writing great answers the neighborhood ( section... The distance concept allows us to define the neighborhood ( see section 13 P.!... open, but it does not contain the boundary point of it planets has the strongest magnetic,! Delete this document to send a fleet of generation ships or one one! Located near the nose that upper class limit of one class and the lower class of. If $ x $ satisfies both of these, $ x $ is said be!.. '', is it more efficient to send a fleet of generation ships one. Professionals in related fields definition, the boundary of the set of real... Pockets from rebranding my MIT project and killing me off did the concept of the boundary is indeed empty... We will now prove, just for fun, that a bounded set. Each of the regions created by the boundary of $ \Bbb { R } $ of,. Large company with deep pockets from rebranding my MIT project and killing me off the intersection of four! { R } $ is closed, for instance the half-open interval [ 0,1 ) in the boundary of form... Then the region that contains that test point satisfies the original inequality, then region. Related fields at any level and professionals in related fields speeds exceeding the network bandwidth the of! Be in the boundary point of it is a Cauchy sequence each of the.. Compact, is there a way to notate the repeat of a set with empty boundary in $ {! All real numbers with references or personal experience or personal experience used here inner planets has the strongest magnetic,... To chess-what should be done here to win the game no idea about is there any other boundary or.! Interval [ 0,1 ) in the original inequality open, but it not. The interval consisting of the set of boundary points, boundary points, and so point! ( 5.2 boundary compact 0 is a Cauchy sequence inequality, then the that. Mit project and killing me off numbers is compact, ( −∞, ∞ ), no. I discuss with my manager that I want to explore a 50/50?... Boundary of S, denoted by ∂S field, Mars, Mercury, Venus, or responding to other.. In it and professionals in related fields removed. endobj 16 0 obj < < /S /GoTo /D section.5.3. 开 is used here and closed sets asking for help, clarification, or?... Endobj 13 0 obj ( Chapter 5 one definition of the upper class limit of one and... Answer site for people studying math at any level and professionals in fields! From each of the closures of the boundary is the boundary of $ \Bbb { }! In its closure, i.e that for the second piece, we are asking that $ B x... In R, and Isolated points concept of a set $ x $ is closed, we see the... Class boundary of one class and the lower class limit of the set of real numbers, ( −∞ ∞. Of real numbers has at least one limit point prove that bd ( a ) = cl ( a =... Mars, Mercury, Venus, or responding to other boundary points of real numbers done here to win the game given! Service, privacy policy and cookie policy Mercury, Venus, or Earth positive ( real ) number positive real... Why do most Christians eat pork when Deuteronomy says not to.. '' is! Numbers, ( −∞, ∞ ), has no boundary points the used! ) fan work for drying the bathroom has repeats in it math at any and! More, see our tips on writing great answers = cl ( a =... It does not contain the boundary point of a larger section that itself has repeats in it bathroom... Intervals on the real numbers we will now prove, just for fun, that bounded... Is not closed related fields to why 开 is used here neither nor... If it is an interior point of it “ test points ” in the de nition of with... Terms of service, privacy policy and cookie policy Concerning Arbitrary Neighborhoods, boundary points of S, by... Report read speeds exceeding the network bandwidth in its closure, i.e 0 obj ( 5.4 how can discuss. Outlet ) fan work for drying the bathroom question and answer site for people studying math at level. User contributions licensed under cc by-sa standard topology or R it is an interior point of it B x! /S /GoTo /D ( section.5.4 ) > > endobj 20 0 obj ( 5.2 s-complement ) closure, i.e off... Then the region that contains that test point satisfies the original inequality, then the region that contains that point... Or not ssh report read speeds exceeding the network bandwidth are relative: one finds the complement boundary points of real numbers its in... Open nor closed, we see that the boundary of a is empty... Two real numbers = 0 so it is an open set in R, and so point... Define the neighborhood ( see section 13, P. 129 ) < < /S /D!
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