8. Show: An infinite set of points need not be a closed set.
7. (5 points) Show that the following set is infinite by placing into a one-to-one correspondence with a proper subset of itself. Show the general terms of the set and subset. (a) [3,6,9,12,15...
complwx analysis
7. Show that the accumulation points of any set form a closed set.
4. Show that a set A is infinite if and only if it is equivalent to a proper subset of itself (Problem 21 on page 43).
30. (Hard) (Fourier series) Show that the infinite set はァsin nz.lcos nz : n-1,2. . . .} is an orthonormal set in the vector space C10,2 equipped with the inner product of real continuous functions on the interval io,2π] 2π
30. (Hard) (Fourier series) Show that the infinite set はァsin nz.lcos nz : n-1,2. . . .} is an orthonormal set in the vector space C10,2 equipped with the inner product of real continuous functions on the interval io,2π] 2π
6.1.3. Consider the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. (Such a matrix has entries aij, where i and j are natural numbers. For each such matrix, there is a natural number n such that aij 0 ifi-n or j 〉 n.) Show that the set of such matrices is a ring without identity element.
6.1.3. Consider the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. (Such...
I know a is removable, but need proof. B is
discontinuous/infinite but need proof. C is infinite but need
proof. They can be proved with definitions or with examples.
For the following, state the set of numbers for which the function is discontinuous. Prove your claim. Classify the type of discontinuity. (25 points) x2 – 1 Vx-1 1 A. y = cot x B. y = COS X C. y = x2 – 2x – 1 3x – 2
Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.
Show using a cross-product construction that the class of regular languages is closed under set difference. You do not need an inductive proof, but you should convincingly explain why your construction works.
1. Prove that for any set S S R, S is closed if and only if Se is open. Notice the book has a proof of this, but it uses a different notation for set complements and a different definition of neighborhood. You may consult it, but you must write your proof using the definition for interior point I presented in class (also in the notes on blackboard). If you copy the proof from the book you will not receive...
6. Let X be an infinite set and let U = {0}U{A CX :X \ A is finite}. (a) Prove that U is a topology on X. (b) Let B be an infinite subset of X. What is the set of limit points (also known as "accumulation points") of B? (c) Let B be a finite subset of X. What is the set of limit points of B?