MATH V1207 Lecture Notes - Lecture 3: Function Composition, Surjective Function, 5,6,7,8

ey table is seen below xeecions tor Chapter 1 12, Fori, b € Ζ.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

42 Metric Spaces Proof. The proof is somewhat lengthy but straightforward. We subdivide it into four steps (a) to (d). We construct (a) X=(X, d) (b) an isometry T of X onto W, where W Then we prove: (c) completeness of X. (d) uniqueness of X, except for isometries. Roughly speaking, our task will be the assignment of suitable limits to Cauchy sequences in X that do not converge. However, we should not introduce "too many" limits, but take into account that certain se- quences "may want to converge with the same limit" since the terms o those sequences "ultimately come arbitrarily close to each other intuitive ide " This a can be expressed mathematically in terms of a suitable equivalence relation [see (1), below). This is not artificial but is suggested by the process of completion of the rational line mentioned at the beginning of the section. The details of the proof are as follows. X-(X, d). (%) and (x) be (a) Construction of Let Cauchy sequences in Х. Define (4) to be equivalente, to (x), written Let X be the set of all equivalence classes,of Cauchy sequences thus obtained wewrite.(%)eito mean that (x.) is a member of i (a representative of the class , we now set where (t)e i and (y)e y. We show that this limit exists. We have hence we obtain and a similar inequality with m and n interchanged. Together, 1e review of the concept of equivalence, see Al 4 in Appendix 1

1.6 Completion of Metric Spaces 43 Since (%) and (%) are Cauchy, we can make the right side as small as we please. This implies that the limit in (2) exists because Ris complete. We must also show that the limit in (2) is independent of the particular choice of representatives. In fact, if ( )-(x) and (%)-(y"), then by (1), as n-a, which implies the assertion We prove that d in (2) is a metric on X. Obviously, d satisfies (MI) in Sec. 1.1 as well as d(i,i)s0 and (M3). Furthermore, gives (M2), and (M4) for d follows from TU ) by letting n-00, with (b) Construction of an isometry T: X-Wc each bex we associate the class bex which contains the constant Cauchy sequence (b, b,). This defines a mapping T X_W onto the subspace W= TX) ,. The mapping T is given by b-→ b= T, where (b, b,.Jeb. We see that T is an isometry since (2) becomes simply メ here é is the class of (ya) where y- c for all n. Any isometry is injective, and T: X-→ W is surjective since T(X)=w. Hence W and X are isometric; cf. Def. 1.6-1(b). We show that W is dense in X. We consider any iex. Let ()e i. For every e>0 there is an N such that in> N)

Let (쯔스-k h. Then This shows that every s-neighborhood of the arbitrary i e X contains an element of W. Hence W is dense in X. (c) Completeness of X. Let () be any Cauchy sequence in X. Since W is dense in X, for every there is a &ne W such that Pl Hence by the triangle inequality and this is less than any given ε>0 for sufficiently large m and n because (辶) is Cauchy. Hence (2-) is Cauchy. Since T: X-w is isometric and i W, the sequence (z), where zm-Tim, is Cauchy in X. Let te X be the class to which (.) belongs. We show that i is the limit of () By (4) rt Since (m)e i (see right before) and i e W, so that (2 the inequality (5) becomes ei nm to and the right side is smaller than any given e>0 for sufficiently large n. Hence the arbitrary Cauchy sequence (L) in X has the limit ieX, and X is complete.

1.6 Completion of Metric Spaces 45 っn+ワ Fig. 11. Notations in part (d) of the proof of Theorem 1.6-2 (d) Uniqueness of X except for isometries. If (X, d) is another complete metric space with a subspace W dense in X and isometric with X, then for any i, fe X we have sequences ( ), (SJ in W such thatn and -y; hence follows from [the inequality being similar to (3)1. Since W is isometric with We X and W-X, the distances on X and X must be the same. Hence X and X are isometric. We shall see in the next two chapters (in particular in 2.3-2, 3.1- and 3.2-3) that this theorem has basic applications to individual incomplete spaces as well as to whole classes of such spaces.

■Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a curve y f(x) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It's often useful to start by determining the domain D of f, that is, the set of values of x for which f(x) is defined. B. Intercepts The y-intercept is f(0) and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set y0 and solve for x. (You can omit this step if the equation is difficult to solve.) -x C. Symmetry (i) If f(-x) =f(x) for all x in D, that is, the equation of the curve is unchanged when x is replaced by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for x0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a)]. Here are some examples: y-x, y x", y-lal, and y cos x. metry (ii) If f(-x) =-f(x) for all t in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y-x,y-x, y = x, and y-sin x. nmetry