Info

Figure 2.14 AU(2,3)l

(ii) Are[(2, 3)J = {(.v. y) e R2 : - 2)2 + (y - 3)2 < e). This is the set of points in R2 lying inside a circle centered on (2. 3), and with radius e. See figure 2.14. _

(iii) yvf[(2,3. D] = {(*. y. z) e R3 : J(x - 2)2 + (y - 3)2 + (z - I)2 < e\, This is the set of points lying within a sphere centered at (2, 3. 1) and with radius t.. Figure 2.15 illustrates this neighborhood. ■  A set .V C R" is open if, for every x e X, there exists an ( such that Nf{x) C .V.

Thus a set is open if it is possible at every point within it to find an e-neighborhood of that point that lies entirely within the set.

A boundary point of a set X C R" is a point xo such that every e-neighborhood Nt(xo) contains points that are in and points that are not in X.

Given these two definitions, it is clear that an open set does not contain any boundary points it may have.

Definition 2.21

A set A' C R" is closed if its complement A C R* is an open set.

2.3 SOME PROPERTIES OF POINT SETS IN H" 39

Definition 2.22

A set X C R" is bounded if, for every ¿o € X. there exists an e < oc such that X C Nt{xn).

In other words, a set is bounded if it can be enclosed in a (sufficiently large) e-neighborhood of any of its points.

To formalize the definition of convexity of a set in R". we need first to gener alize the idea of a convex combination.

Definition 2.23

Given two points x,x' e R". their convex combination is the set of points x e for some A e [0. 11, given by x = kx + (I - A).v' = [AAI -1- (1 - k)x\ kxn + (1 - k)x'n]

Example 2.15 Find the convex combinations of die points

Solution

(i) x = A(2, l) + (l - A)(—3.2) = 12/.-3(1 - X), k + 2(1 - A.)]. For example, if A = 1/2, JE = (-1/2.3/2).

(ii) X- = A(,2, 1.0) + (1 - A)(—3. 2. I) = [2k - 3(1 - A), A -I- 2(1 - A), 0 + (1 - A)|. For example, if A = 1/4. x = (-7/4,7/4.3/4).

(iii) Jc = A(2, 1,0,-2) + (I - A)(-3,2, 1,5) Or x = [2A - 3(1 - A),A + 2(1 - A). 0 + (1 - A). -2A + 5(1 - A)]. For example, if A = 2/3. x = (1/3, 4/3. 1/3,1/3). ■

In definition 2.23, we have implicitly introduced the idea of adding points in R" by adding their corresponding coordinates. This is a first step in developing linear algebra, which we will take up in detail in chapters 7 Lo 10.

Intuitively in R2 and RJ the convex combination of two points consists of the set of points lying on the line segment between those points, as figures 2.16 and 2.17 illustrate.