→ Det f : V → Il Xa be defined as : a e A f (v) = (fa(v)) a e A where fa : V ¸Xa for each a e A. Let II Xa have the product a e A Topology, then the function f is continuous if and only if each function fa is continuous.

Q: Suppose that f: A → B is a function, C ≤ B and D C B. Prove that ƒ¯¹(C − D) = ƒ¯¹(C) − ƒ¯¹(D) -

A:

Q: Let f: A→ X X Y be defined as f (a) = (f₁ (a). f2 (a)) each a ≤ A, Then f is continuous if and only…

A:

Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

A:

Q: Let f: XY and g: Y→ V be functions. Show the following: (a) f is injective 3h: Y→ X such that hof=…

A:

Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

A:

Q: Let L: R→ R be a linear operator, if L(X)= L(Y), then X= Y. Select one: O True O False

A: Given Linear operator is  L:R3→R3 we have to check if L(X)=L(Y) the X=Y

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:

Q: Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...)…

A: Here, in the question, there are given the bounded linear operators l1→l∞ which are given as…

Q: Show that the function f: C → C given by f(z) = z2 is continuous on C.

A:

Q: Let f : X → Y and g : X → Z be two bijective functions. Is the function h: X → Y × Z defined by h(x)…

A: Given: f:X→Y and  g:Y→Z be two bijective functions. To Check: Whether the function h:X→Y×Z defined…

Q: Prove that the function f: R → R: x x² + ax + b where a, b E R is continuous at every c E R.

A:

Q: If a function f: D R is differentiable at a point (a, b)ED, then f is continuous at (a, b);

A:

Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

A:

Q: Let G = &lt;z | z6 = 1l and H = &lt;x, y | x2 = y3 =1, xy =yx&gt;. Showthat G and H are isomorphic.

A: Given :  G = z : z6 = 1 and  H = x, y : x2 = y3 = 1, xy = yx To show : G and H are isomorphic.

Q: Let f : R → R be a function given by f(x) = 2x – 1 (a) Determine whether f is bijective or not. (b)…

A:

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: b) Let f: D→ C be a function and Y a subset of C. Show that f¹ [C-Y] =D-f¹ [Y].

A:

Q: X1 X1 Let L: R R be a linear operator such that L X2 )=X1 then Im(L) spanned by: X3 X1 Select one:…

A:

Q: 4. Define the operators T and U on the space P(R) of all polynomials in variable x as follows:…

A:

Q: A = [a,b), be an (right half-open) interval of R. 6 (c) (d) a,be R+,b+00, Explain whether f(A) is…

A:

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…

A:

Q: 18. Let A = {1,2, 3}, B = f: A → B, and g : B → C, such that {W, Y, Z}, C = {7, V2, V5}. Find…

A:

Q: Forz € C, define the linear operator T₂ : C → C by T₂(w) = W. Characterize those z for which Tz,…

A:

Q: Define the relation A=((x,x+ 1):xER] Show that A, is a function from R to R.

A:

Q: The function f is defined as follows: f: R² R² with f(x, y) = (2x, x+y) Which statement is correct?…

A:

Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

A:

Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

A:

Q: Rocky Mountain Chocolate Factory (RMCF) founder and president Frank Crail employs 220 people in 361…

A: for net price  equivalent rate determine first find the complement of the chain discount rate…

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: