い 3) (= {00000, 11 000,00 111, 111113 Find u sit. C is exactly U-error-detecting. Cis Find v s.t. C is exactly V-error- correcting

Q: iii) True or False: Every subspace of a complete metric space is complete.

A:

Q: i) True or False: T = {0, R} U {(-o, a]: a € R} is a topology on R.

A:

Q: Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {6, X, {a}, {a,…

A: I

Q: Q3 Give an example of a connected subspace of R2 which is not path- connected.

A:

Q: Q1. Let X = {m,n, p, q}, {ø, X, {p}, {m, n}, {p, q}, {m, n,p}, {q}} and Y = {a, b, c}, F =…

A:

Q: Let I be the set of irrational numbers with the usual topology (that is, the topology it inherits as…

A: In topology, a component of a topological space is a maximal connected subset of the space. In other…

Q: d if -1<r e if r<1 ist the open sets in the quotient topolo

A:

Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: Consider the equivalence relation on R defined by r ~ y if x – y € Z. Describe the quotient space…

A: In topology and related areas of mathematics, the quotient space of a topological space under a…

Q: Q4. Let (X,T) be a topological space where X = {m,n, q, r} and T = {4, X, {q}, {r}, {q,r}, {m, q,…

A:

Q: Lef p(m)= 1+ x be iredeucrble polymomial in F [n]. an O Construct field uith 16 elements .

A: Given that : px=1+x3+x4​is irreducible polynomialField = F2xNow construct the field with 16 elements

Q: 4) Let 7 be the Euclidean topology on R and 71 be the discrete topology on R. Define f: (R, 71) →…

A:

Q: X be given by J(*) 2. Let X be the set of real numbers, let U be the usual topology, and let D be…

A: (a).         f : (X, D)  to  (X, U)   is continuous. Because if we take any set open in (X, U) ,…

Q: Let X be the set of reals, and let T = {SCX:0€ X-S} U {X}. Show that T is a topology for X and find…

A:

Q: Find recursive definitions and closed formulas for the arithmetic sequences below. Assume the first…

A: Given,2,5,8,11,14Here a0=2d=a1-a0d=5-2d=3we know that ,an=a0+(n-1)dan=2+(n-1)3an=2+3n-3an=3n-1

Q: Match the homeomorphic figures. ....F Prove or disprove the following: FA and R are pathwise…

A: Given:- (a)Homeomorphic figures     △.................. F  T.................. 0∞...................…

Q: For z # 5i, let g(2) = (2 – 5i)Log(z – 5i) + (z – 5i) where Log is the principal logarithm. Find a…

A: The given function is g(z)=(z-5i)Log(z-5i)+(z-5i) That is z0=0 We have to find the power series of…

Q: Let X be an infinite set with the countable closed topology T-{S subset of X; X-S is countable}.…

A: Given that X is an infinite set with the countable closed topology T-S subset of X; X-S is…

Q: f(x) = {x + 2 if x > 1 (2x – 1 if x < 1 then f -'( ]2,4[ ) is then f-(]2,4[) is [1,2[ and f is not…

A:

Q: Show that any infinite subset A of a discrete topological space X is not compact.

A:

Q: Find lower bound for f(n) = 5n².

A: To find the lower bound for f(n) = 5n2

Q: Let (X, T) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

A:

Q: Which of the following sets is open in (IR, Tu)? OR-(Q-Z) (1)"2n? nin?:nE N} R - [2,5) O 12, 5) U…

A:

Q: Show that the class of closed interval [a,b), where a is rational and b is irrationals and a <b, is…

A: Base for topology: A collection B of subsets of topological space X is said to be base for topology…

Q: 3.1 Terminology 1. Probability is a measure of the chance (likelyhood) of certain results (outcomes)…

A: Probability gives the measure of the chance of happening an event. Fill the incomplete blanks.

Q: Prove that every finite T,- space has the discrete topology.

A: