(b) Let U, T be usual and cofinite topologies on se:t of real numbers R and f be a map from R into R such that f(x)=1x-21 for any xER, show t.hat wheter f is U-T continuous or not and t--U continuous or not. Such that

Q: 7. (10 points) Let f: A B be onto. Prove that S((C)) = C for all sets CC B.

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Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

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Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: Let S be set of all strings of a's and b's. Define a relation R on S as follows: Vs, t E S, sRt e…

A: Let A be any and let R be any binary relation on the set A. Relation R is symmetric if for a, b∈R,…

Q: Suppose f and g are continuous complex functions on a metric space X. Prove f + g is continuous on X

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Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: consider the space generated by the set of functions {cos nx , sen nx} with ,n = 0,1,2, … show that…

A: We will prove  the given statement.

Q: Let A = {1, 2, 3, 4, 6}. Let R be therelation on A defined by {(a, b): a , b ∈A, b is exactly…

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Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

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Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: Suppose the function f: X --> Y is onto. Prove or disprove that the induced map f-1: P(Y) --> P(X)…

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Q: A function, F: R × R ---› R x R has been defined as follows: F(x, y) = (3y - 1, 1 - x) for all (x,…

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Q: Let V = span{e2", xe2", x²e?=}. (a) Show that (a1e2a + a2xe2" + a3x²e2") e V for any a1, a2, az E R,…

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Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

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Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: Determine whether the relation R on the set Z is reflexive, symmetric, antisym- metric, and/or…

A: Given: The relation ℝ on the set ℤ and x,y∈ℝWe need to check whether the given ℝ on the set ℤ is…

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: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: Let f: R2 R2 be defined f(x, y) = (3x + y,x+y). (a) Prove that f is a bijection. (b) Keep f the same…

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Q: Let A = {1,2,3}, X = {1,3}, and R is a relation on A, such that R= {(x,y) | x<y ^ x, y = A} R"X =…

A: , and is a relation on .

Q: Construct a one-to-on continuous mapping of the set {(#1, 12, - ..., In); 0 < T; < 1. j= 1, 2,…

A: Given that : x1,x2,x3,.........,xn, 0&lt;xj&lt;1, j=1,2,3,.....,n So we have to take ,…

Q: Which functions are one-to-one? Which functions are onto? Describe the inverse function for any…

A: One to one: A function f : A → B is one-to-one if for each b ∈ B there is at most one a ∈ A with…

Q: on 1: a) Let O be the set of odd numbers and O' = {1,5, 9, 13, 17, ...} be its subset. Define the…

A: Since the part b is independent of the first part so I am answering the first question only for the…

Q: B Let A = {(x,y) |x-yeQ for x, y = R} be a relation on K. Which statement best describes problem?…

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Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

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Q: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis

Q: Let ƒ be a mapping defined by f : x + iy → -Y x Show that f(2122) = f(21)f(22) V 21, 22 E C.

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Q: Prove that f is contraction map if and only if (X,τd)  to ( Y,τc)  is continuous

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