Q. Determine whether the given set V is closed under the operations addition and multiplication where V is the set of all ordered pairs of real numbers (x,y) where x > 0 and y > 0 (x, y) + (x' + y') = (x + x',y + y') And с. (х, у) 3 (сх, су)

Q: Sketch the set S = {z:|2iz – 2 – 5i| > 2} and state clearly if it is open/closed/neither,…

A: Solution: The set is S=z:2iz-2-5i≥2

Q: Prove that (x - i) = > xi i=1

A: We will prove the following result.

Q: (1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

A: First let xy be positive  Then either both x and y are positive or both are negative  If both are…

Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

Q: 3) Let A be a nonempty set of real numbers that is bounded below. Let -A be the set of all numbers…

A:

Q: (Do solution short form)Prove that the only idempotent elements in an integral domain R with unity…

A:

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

A:

Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

Q: Prove that ', 12 and lº norms satisfy the properties (N1)-(N3). (N1) (Positivity) ||x|| > 0, and ||x…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let 'H' and 'Gʻ are two non-empty sets which are defined as; H = (x; -1 sIs1}=G and modulus function…

A:

Q: Prove that the only idempotent elements in an integral domain R with unity are 0 and 1.

A: Idempotent element : An element a in a ring R is said to be an idempotent element if a.a = a .…

Q: consider the space generated with the set of functions  {x, x^2} , show that the set of elements of…

A:

Q: If X is inner product space over real numbers then the inner product always greater than or equal…

A: The objective is to determine whether the given statement is true or false. Given statement is, " If…

Q: 4. Show that both functions x and sin(x) belong to inner product in this space. L² ([0, 1]) and find…

A: L20,1 is a vector space of all functions f:0,1→ℝ satisfying ∫01f(x)2dx<∞. And the inner product…

Q: Prove or disprove that ⌈x + y⌉ = ⌈x⌉ + ⌈y⌉ for all real numbers x and y.

A: According to the given information, It is required to prove or disprove that:

Q: Write one or more inequalities that describe the set of points. The closed region bounded by the…

A: For closed region , Sign of equality holds. Equation of sphere is -  x2+y2+z2=a2 ,where a is the…

Q: Let Q be the set of all rational .numbers a) If T= {0}U{ACQ: Q-A isfinite} prove that (Q, 7) is…

A: For the solution follow the next steps.

Q: f for ƒ : R - {2} → R- { 5} indicated by f (x) =(5x +1)/(x -2a), a bijection can be made from [0, 1)…

A: Bijective function: The domain is set A and the co-domain in set B relates elements of two sets A…

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: For the points (x1, y1) and (x2, y2) in the plane R?let (x1, y1)~ (x2,Y2) mean that xỉ + yỉ = x + y…

A:

Q: (7.8) Lemma. Let K, L be simplicial complexes which intersect in a common subcomplex, then z(K U L)…

A:

Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

A:

Q: hat subring R[x², x°] not isomorphic to R[x].

A:

Q: Lie: Prove that every finite set S of R" has no accumulation point.

A: We have to prove this given statement

Q: Let x be a positive integer and let R = {a1, a2, a3, . . . , ax} be a nonempty set where ai are real…

A:

Q: prove that [x+y] > [x] + [y] For all real number x and y

A: Let,  x=n+a,y=m+b where n,m are integers and a,b are fractions such that 0<a<1,0<b<1…

Q: 2- Find the set of lower bounds and the set of upper bounds of the following sets in Q, if it…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1- Find the set of lower bounds and the set of upper bounds of the following sets in R, if it…

A: Since you have posted multiple sunparts of a question so according to our guidelines we will solve…

Q: Prove that the equalities in the given question hold for all xy, , z, and winZ . Assume only the…

A: To prove, x·0=0

Q: 5. Verify that if U is open in X and A is closed in X, then U-A is open in X, and A- U is closed in…

A: We have given that , U is open in X. A is closed in X. We need to show that , U - A open in X. Also…

Q: A2

A: Consider the matrix A is, A=0100 Apply the condition, A2 = O. 0100·0100=0000 The matrix A is…

Q: T-space X has no Question 1. Show that a finite subset of accumulation points.

A:

Q: 4. Let X1, ,X4 be iid Unif(0, 0). (a) Show that X(4) is independent of X(1)/X(4). (b) Show that X(4)…

A: Given that X1,X2.............,X4 be iid Unif0,θ (a) X4 is complete sufficient statistics and…

Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Use Eisenstein Criterion to verify that , If p = 2, f(x) = x² – 2 is irreducible %3D over Q. State…

A: Explanation of the solution is given below...

Q: 8. Determine whether v(x) = r'+3|x| even, odd or neither. Discuss its symmetry.

A: Q8) Given: v(x)=x3+3x

Q: 2. Let a,, az,.., a, be numbers. Show that there is a single number r such that foxo: (-a,) + (z –…

A: Solving

Q: Prove or disprove that [x + y] = [x] + [y] for all real numbers x and y.

A: we have to prove or disprove that [x+y]=[x]+[y] for all real numbers x and y.

Q: A set S is called 'denumerable' if there exists a bijection f : N → S. (a) Show that the set N>2 is…

A:

Q: show that the set [1, cos x, cos 2x, … }is orthogonal on the interval [−pi, pi] and determine its…

A: First we have to show that the set 1, cosx, cos2x, …… is orthogonal on the interval -π, π. Let ϕ0x=1…

Q: Let x, y and z be real numbers. If x > y and y = z, then x > z. For any real numbers x and y, with x…

A:

Q: Let n be a positive integer and let R = {a1, a2, a3, ... , an} be a nonempty set where a; are real…

A: GIVEN: R = {a1, a2, a3,........, an}

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: If fi and fa are two bounded and integrable funct ions on [a, b] and there exists a number >0 such…

A:

Q: O Principal Ideal Domain

A: Using theorem - Polynomial ring R[x] is pricpal ideal domain iff ring R is field itself Now given D…

Q: a) Check that the functions e^x, e^-x,coshx are solutions of y'''-y'=0 b) show directly that…

A:

Q: Find the field of quotients of integeral domain z[i] and z[√2]

A: We will solve this question using basics of ring theory and just the definition of Fields and…

Q: A sequence (a1, b1), (@2, b2), (a3, b3), -· - - of points in the co- ordinate plane satisfies (am+1;…

A: A sequence a1,b1,a2,b2,a3,b3,... of points in the coordinate plane satisfies the equation…