(7) List all isomorphisms f : Z2 × Z2 → Z2. List all isomorphisms (7) List all isomorphisms f : Z2 × Z2 → Z2. List all isomorphisms
=n. 1: (a) Let n = 2020. List all of the pairs (x, y) E Z2 such that x2 - y2 (b) Let n = (39). Find the number of pairs (x,y) e Zé such that x2 + y2 = n2. (c) Write 650 as a product of irreducible elements in Z[i], then list all of the pairs (x, y) e Z2 with 0 < x <y such that x2 + y2 = 650.
+3 7 اذا كانت الدالة - 22( z2 + 2) = (2)f فان (0,Res ( f هو ه م 2 م لا | -
1. Give a complete list of all numbers a for which z2 +1 > ar. 2. Definition: A function f is even if f(-x) = f(x) for all inputs z. A function f is odd if f(-x) = -f(x). (a) Let f be any function with domain (-0,0). i. Show that the function g(x) = f(x) + f(-x) is even. ii. Show that the function h(x) = f(0) - f(-x) is odd. iii. Show that f can be written as...
. Let F(x, y, z) = ze+7 +(+ Iny)ī – (z2 + arctan y)k. (a) Calculate the curl of F, or 7 x 7. (b) Calculate the divergence of F, or VF.
Q.7, as question above
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
7. Let S be surface, oriented upward, given by the graph of f(x, y) 2 - z2 -y2 which lies over the unit disk in the ry-plane. If F(, y,z (-4x+ 1 + 3y2 1 +3y2
7. Let S be surface, oriented upward, given by the graph of f(x, y) 2 - z2 -y2 which lies over the unit disk in the ry-plane. If F(, y,z (-4x+ 1 + 3y2 1 +3y2
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
I. Functions and Isomorphisms. Let G be a group and let a EG be any non-identity element (so a #e). Define a function f : GG so that, for any r EG, f(x) = (xa)-1 (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. (c) Is f an isomorphism? Prove your answer.
Let f=x3 +x2 + x + 1 in Z2 [x]. Write f as a product of irreducible polynomials in Z2 .
(1 point) Calculate ſls f(x, y, z)ds For y = 4 – z2, Is $(x, y, z) ds = 0 < x, z <7; f(x, y, z) = z