Consider R^2 with its standard topology. True/False (justify): There exists a nonempty proper subset A of R^2 such that Bd(A) = ∅.
Consider R^2 with its standard topology. True/False (justify): There exists a nonempty proper subset A of...
Question 13 (1 point) For any bounded and nonempty subset X CR, we have sup(-2X) = (-2) sup X. Here -X := {r ER : -TEX} O True O False
topology
Consider The The for and radius 1 in see R² with felloeding secrets delay) : _ 14-442+ IV-vel doo (ny) = max { 10 -4 2/V-val} XELU, 4) and y = 1 U2 V22 a) Sketch open ball Billo centered al (1,1) both (R3 d.) and (IR² doo) prove That if u is open subset of (R², do it is also an open subset of (R² doo) @ Also -Prove that if u is open subset (R², do then...
In the following exercises, consider the metric space R with the discrete metric and the subset A [0, 1 C R. 15) True/False: A is closed. 16) True/False: A is open 17) True/False: Every point of A is a limit point of A. 18) Calculate the boundary of A. 19) True/False: For all Xo E X and all ε > 0, if B(x0, e) contains a point of A besides xo, then A C B(xo, e)
In the following exercises,...
Problem 5. A subset A C R is an affine subspace of R" if there exists a vector bE R" and an underlying vector subspace W of R" such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2 (b) Consider A E Rnx, bER" and the system of linear equations AT . Prove that: (i) if Ais consistent, then its solution set is an affine subspace of R" with underlying (ii) if At...
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y of R such that (Y, dy) contains nonempty closed and bounded subsets which are not compact (here dy is the metric inherited from the usual metric in R)
Using only the definition of compact sets in a metric space, give examples...
Only 5-9 please
1. (10 points) True/False. Briefly justify your answer for each statement. 1) Any subset of a decidable set is decidable 2) Any subset of a regular language is decidable 3) Any regular language is decidable 4) Any decidable set is context-free 5) There is a recognizable but not decidable language 6) Recognizable sets are closed under complement. 7) Decidable sets are closed under complement. 8) Recognizable sets are closed under union 9) Decidable sets are closed under...
True or false. Please justify
why true or why false also
(I) A square matrix with the characteristic polynomial 14 – 413 +212 – +3 is invertible. [ 23] (II) Matrix in Z5 has two distinct eigenvalues. 1 4 (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices D1 and D2 are similar if...
Prove that the open balls (disks) in R^2 form a basis for the standard (product) topology.
Determine whether each statement is True or False. Justify each answer. a. A vector is any element of a vector space. Is this statement true or false? O A. True by the definition of a vector space O B. False; not all vectors are elements of a vector space. O C. False; a vector space is any element of a vector. b. If u is a vector in a vector space V, then (-1) is the same as the negative...
True or false, explain if false please
(a) Given a matrix, the dimension of its kernel plus the dimension of its image is the number of its rows. (e) The vectors 1, t, t are a mutually orthogonal subset of C(0,1] with the standard Linner product.