Let B and B* be bases for topologies T and T*, respectively, on a set X. Suppose that each B EB is the union of elements of B*. Show that T is coarser than T* (i.e., TCT*).

Q: 7. If V=set of integers ending with 3; then V is O C a. A vector space under addition multiplication…

A:

Q: Let B1 be a basis for a topology T on a set X and let B2 be a collection of open sets containing B..…

A:

Q: Let S CV be a subset of a vector space. Recall that the span of S is: Span(S) = {a1v₁ + a₂V₂ + +…

A:

Q: 1. Give an example of a set S with a relation ~ on S such that ~ is symmetric and transitive but not…

A:

Q: Find a base for the Standord tapology-

A: This is a question of topology.

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

A:

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

A:

Q: A. Let R be a relation on set A. A={x :xeJ and 0 sx s 12), and R={(a,b) : la - b| is multiple of 4}.…

A: Since not particular question asked, as per guidelines solution to only first question(i.e A) is…

Q: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

Q: Let S be a set of four vectors in R* with the property that every subset of S consisting of three…

A: The solution are next step

Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

A:

Q: Determine a sine equation for the graph shown below. Justify your answer. 4 2 0 OL M

A: Consider the graph of given function.We need to find the sine equation of the above graph.

Q: A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12}, and R={(a,b) : |a – b| is multiple of…

A:

Q: Any two bases contain the same number of vectors. Let dim V = n. If you have n linearly independent…

A: Assume we have two different bases for Since B1 is a basis, every vector in B2 can be expressed as a…

Q: 2.44 Show that if a vector is in the span of a set then adding that vector to the set won't make the…

A: The given statement is " if a vector is in the spanning set then adding that vector to the set would…

Q: 1. Let {v₁,..., Vn} be a linearly dependent set. Prove that there exists some j with 1 ≤ j<n such…

A: Suppose that    be a  linearly  dependentset.Then there exists some   such that where  are  scalars…

Q: A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12},  and R={(a,b) :  a - b  is multiple…

A: Since not particular question asked ,as per guidelines solution to only first question (i.e A) is…

Q: If A is a bounded linear operator on a H, if Hilbert space H, prove that R(A*) = and only if A is…

A:

Q: Show that the o-algebra B of Borel sets contains all F,-sets.

A:

Q: Find the null space for A = 1 2 What is null(A)? {(} O A. O B. R? {} O C. span O D. span: 2 2. O E.…

A: We have to find the nulllspace of the given matrix .

Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

A:

Q: 2.54 For each n e N, let X„ be the set of polynomials of degree equal to n, and let Pn = U-o Xk.…

A: Sol

Q: Consider the measurable space (2, F), and let D be a sub o-algebra. Let X and Y be D-measurable…

A: The objective of this question is to prove that aX and X + Y are D-measurable random variables,…

Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: Let A be a nonempty subset of R that is bounded below. Define the set B = {b: b is a lower bound of…

A:

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Suppose A=[v_1...v_n]. Prove that if A(x)=b is consistent if and only if b is an element of…

A:

Q: Determine which of the four inner product axioms do not hold. Give a specific example in each case.

A: Given that, u=u1u2 and v=v1v2 in R2 Giving inner product u·v=u1v1-u2v2 Axiom (4) of inner product…

Q: Let < be a partially order on a set A. Show that there exists a linear order 3 on A such that if a…

A: Given that ≤ is a partial order on a set A. The objective is to show that there exists a linear…

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

A:

Q: Let X = (6,7), and d be the ususal mortric, d(r, y) = |r - yl Gire ezample of a set ACX, to be not…

A:

Q: Kailashben

A: The objective of the question is to find a relation R on the set S = {a, b, c} that is not…

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

A:

Q: Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets.…

A: We have a statement  Let A, B and C be sets. Then (A - B) U C = (A U B U C) - (A ∩ B).

Q: Let X be a set with an operation *:X² → X such that (i) x * x = for all x E X, %3D (ii) x * (y * z)…

A: