By Frederick H. Soon
This booklet is meant as a consultant for college students utilizing the textual content, Calculus III through Jerrold Marsden and Alan Weinstein. it can be specified between such publications in that it was once written through a scholar person of the textual content. for every portion of the textual content, the advisor features a record of necessities, a evaluation quiz (with answers), a listing of analysis pursuits, a few tricks for examine, recommendations to the odd-numbered difficulties, and a quiz at the part (with answers). for every evaluation part the consultant contains ideas to the atypical routines and a bankruptcy try with options.
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Additional resources for Student's guide to Calculus by Marsden and Weinstein
4) The union of a finite or countable family of finite or countable sets is finite or countable. Indeed, let (Xi)iEI be such a family, where I is finite or countable like the Xi' For each i choose a surjective map Ii : N -----t Xi and define a map f of the cartesian product N x I onto the union X of the Xi as follows: f((n,i)) = hen) for n E N and i E I. For every x E X there exists an i E I such that x E Xi, so an n E N such that x = fi (n). The map f is thus surjective, and since the product N x I is countable, so is X finite or countable too.
In binary counting, every real number between 0 and 1 can be written with the aid of a sequence of digits 0 and 1, and in a unique way, if one insists that the sequence does not consist only of 0 from a certain point on. If one considers only those digits equal to 1, this amounts to writing x in the form _ (1) PI ( 1 ) PI +P2 ( 1) x- + + 222 PI +P2 +P3 + ... with well-determined integers PI, P2, P3, ... 01111 ... 1000 ... ), one has PI = 2, Pn = 1 for all n > 1. Conversely, such a sequence of integers defines a number between 0 and 1.
From the preceding result, X - D contains a countable set D' and one has X =Y U (D U D'), X - D =Y U D' where Y = X -(DUD') is disjoint from D and D'. It is not hard to construct a bijection 9 of Y onto itself; since D and D' are countable, so is DUD'; thus there is also a bijection h of DUD' onto D'. Then one obtains a bijection f of X onto X - D by putting f(x) = g(x) [for example f(x) = x] for all x E Y and f(x) = hex) for all xED U D'; f is clearly injective and f(X) = flY U (D U D')] = fey) U feD U D') =Y U D' = X-D.