Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x < 0 x = 0 b, x > 0 а, f(x) = с, Is X with the quotient topology Hausdorff? Justify your answer.

Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

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Q: Let A = {1, 2, 3) and let f : A A, defined by f = (1, 2), (2, 3), (3, 1)}. Find f, if it exists. %3D

A: given A = {1, 2, 3} this is function

Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Let F = (y – x, ry) and let C be the line segment from (2, –3) to (1, –1). %3D Evaluate F. dr.

A: We wil first equation of parametric equation of line segment

Q: (-х+а, х < —1 If f(x) = {x² + 5, –1 < x < 2 is continuous then х — b, х 2 2 a + b =?

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Q: Let f: R³ R be the function defined as f(x, y, z)= √√₂² + y² + 2², 0. if (x, y, z) = Q³, otherwise.…

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Q: If S is a subset of T , then u (T) sp" (S)

A: It is Always True

Q: Let A = ū= [¹²3] a [¹3] and v u= LO [2] and define T: R² → R² by T(x) = Ax. Find the images under…

A: The given matrix  and the linear transformation .The aim is to find the images of  and .

Q: Let f: S¹→ S¹ be a continuous function. Proof or disprove the following statements : a) f has a…

A: Given that the  function f : S1 → S1 be a continuous function  where S1 = x,y∈ℝ2 : x2 + y2  =1 We…

Q: Determine whether the given series converges or diverges. If it converges, find its sun 8 Ž(¹+4)* a.…

A: Given series∑n=1∞(1+8n)n

Q: -2y 2x (x²+y²) ²³ (x²+y²)² Compute fF.dr, where F(x, y) = oriented counterclockwise. (Use symbolic…

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Q: Which one of the following relations is not a function? O ((x.y): r, y ER and r + y² = 1) O ((x.y):…

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Q: 4. Show whether or not f is bijective. Letf : R → R_be defined as f(x) = I피+1 %3D

A: A function f is said to bijective if it is one-one and onto

Q: sst. Let f(x, y, z) = x³ yª + zª and x = s5, y = st³, and z = s

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Q: R(s, t) = G(u(s, t), v(s, t)), where G, u, and vare differentiable, and the following applies. v(7,…

A: Given that

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: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

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Q: The largest set on which the function f(x, y) = 1/(2-x² - y2) is continuous is A. The exterior of…

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Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

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Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Consider the function f : [−2, 6] → R, x→ f(x) = 2x3/ [√(x2+8)+2]. i) provide argument why this…

A: This is the question of Real topology, and real analysis.

Q: 1. Show that the following maping f are linear: f : R² → R² defined by f(x, y) = (x+ y,x).

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Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

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Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: Q1\ A 200°F cup of tea left in 65°F room. At time initially the tea is cooling at 5 °F per minute.…

A: The rate of cooling is directly proportional to the rate of temperature difference between the…

Q: We consider a function f from R to R such that f(0)=5. Show that f cannot be a linear map. Hint:…

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Q: of the following statements I. Let f and g be continuous real variable functions defined over the…

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Q: 1.) Which of the following are TRUE about differentiable function: I If rx) exists, then the…

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Q: Let X₁ = [1 3 −1 1]²‚×₂ = [3 −1 3 1]′, and x3 = [-1 2 −3 1]. Is b = T an element of span{X₁, X₂,…

A: No, we can write it.

Q: Consider the relation x2z2 = arctan (xyz) + 4, which defines z as an implicit function of x and y,…

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Q: Let A = {1, 2, 3} and let f : A→ A, defined by f = (1, 2), (2, 3), (3, 1). Find f, if it exists. %3D

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Q: 2. Consider the following three functions: f : Z+ x Z+ → Z+ f(m, n) = mn h: Rx Z+ → Z+ h(r, y) = 7y…

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