Suppose that f :X + Y is a surjection and let yo e Y. Define Z = X -f({yo}) (a) Show that the function g: 2 + Y - {yo}, given by g(x) = f(x) for xe Z, well-defined function. (b) Show that g is a surjection. That denotes
Suppose that f:D+Tis a surjection and let to € T. Define Y =Z-F {{to}) CZ. (1) Show that the function g:Y+T - {to}, given by g(t)= f(t) for t e Y, is a well- defined function. (2) Show that g is a surjection.
How do I prove this function is not surjective? 3.) Let f: R-R, f(x)-x2+ x+1 and Show that f is not injective and not surjective Justify that g is bijective and find gt. PIR, Show all the wortky) Not Surtechive: fx) RB Surjective: ye(o,oo) hng (g) 8 gon)-es is bijecelive g(x)-ex+s
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please find (2y+ye 5. С:/(r)-(cost.sin 1,1). Osis". dy b. Please to prove that F is a conservative vector field: ye". c. Please find J2xye d. Please find the potential function fx, y, z) such that F Vf e. Use the part (d) to evaluate F dr along the given curve C. f. Please find curlF g. Please find curlF Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
(e) Given the functions x 4y 4 Z show that: (i) if both f and g are injective then the composite gof is also injective. (ii) if both f and g are surjective then the composite gof is also surjective. ii) if both f and g are bijective then the composite gof is also bijective. (e) Given the functions x 4y 4 Z show that: (i) if both f and g are injective then the composite gof is also injective....
b) Consider the surface in R3 described by f(x,y,z) = 2x²y3 + z + ye*2 = 9 (i) Find Vf(x,y,z). [3 marks] (ii) Verify that (2,1,0) is a point on this surface. Find the cartesian equation of the tangent plane to this surface at the point (2.1,0). [5 marks]
If 3.80 fig: [a,b] → R 2 Alonspiciens differentiable functions and we suppose Fca) = f(b) =. The wronskien of these a functions is the function TW Cf. g): [a, b] R defined by wCfg) () = det (FX) 906) -F68) g'(x)=9(x)}f'(X) (f'(x) g(x)) If W (f, g) (x) #0 for all x E [a,b], show that it exist a c E Ca,b) such that g (c) = 0.
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
a) A vector field F is called incompressible if div F = 0. Show that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is incompressible. b) Suppose that S is a closed surface (a boundary of a solid in three dimensional space) and that F is an incompressible vector field. Show that the flux of F through S is 0. c)Show that if f and g are defined on R3 and C is a closed curve in R3 then...