Exercise 1.3. Let F be any field. Verify that the following arithmetic properties hold in F....
number a and b 70 Score: B. Bader collin Alhusni_ DATE Вт output E LUBODA RAO+O D ot x 1= X X-D=0 1. X+0=X XXX XX'=0 2. x + 1= 1 Idempotent laws: 3. X+X=X Involution law: 4. (X'=X Laws of complementarity: 5. X+X' = 1 Commutative laws: 6. X + Y=Y+X Associative laws: 7. (X+Y) +2=X+ (Y+Z) =X+Y+Z XY YX (XY)Z = X(YZ) = XYZ Distributive laws: 8. XIY + Z) = XY + XZ De Morgan's laws 9....
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a 3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...
11.1) a) Verify that the function f(x,y) given below is a joint density function for r and y: ſ4.ty if 0 <r<1, 0 <y<1 f(x, y) = { 10 otherwise b) For the probability density function above, find the probability that r is greater than 1/2 and y is less than 1/3. 11.2) For the same probability density function f(x,y) as from Problem #1. Find the expected values of r and y. 11.3) a) Let R= [0,5] x [0,2]. For...
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
Sketch the graph of a function f where all the following properties hold. For full marks, clearly and carefully label all intercepts, relative extrema, inflection points, and asymptotes. • Domain: (-0,00) . Continuous everywhere • Differentiable everywhere except at x = -3 • f(0) = 6 • lim f(x) = 0 • f'(-2) = f'(0) = 0 • f'(x) <0 on (-0, -3) and (0,0) • f'(x) > 0 on (-3,-2) and (-2,0) lim 1' (x) = and lim f'(x)...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
Problem (10 marks) Verify the Divergence Theorem for the vector field F(x, y, z) = (y,1,-) on the region E bounded by the planes y + : = 2 := 0 and the cylinder r +y = 1. Surface Integral: 6 marks) Triple Integral: (4 marks)
Please do exercise 129: Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
1) Let f(3) = 22 (Figure 24). Verify the following: 2-1 (a) f(0) is a local max and f(2) a local min. (b) f is concave down on (-0,1) and concave up on (1,0). (c) lim f(a) = - and lim f2) = 0 -1- 1+ (d) y = x + 1 is a slant asymptote of f(x) as to (e) The slant asymptote lies above the graph of f(s) for x < 1 and below the graph for 2...