Prove for any two sets, E and F, E∪F =E∪(Ec ∩F) Be sure to justify every statement you make by referring back to your definitions.
Prove that Z [i] satisfies the definition of Euclidean Domain : W/Z = N(W) LN(z)
a) Let z,w ∈ C, prove or disprove: Ln(z/w) = Lnz − Lnw b) Find all values in C and the principal value of j^j and ln(-3) c) Find all z ∈ C such that i. tanh z = 2 ii. e^z = 0 iii. Ln(Ln(z)) = −jπ
Let z and w be non-zero complex numbers such that zw /=1. Prove that if z= z^(-1) and w=w^(-1),then (z + w)/(1+ zw) is real.I know z * z^(-1) = 1.
(3) If z = a + ib E C and |2| := Va² + b², prove that |zw| = |z||w]. Proof. Proof here. goes (4) Let y : C× → R* be defined by 9(z) = |z|. Use Problem (3) to prove that y is a homomorphism. Proof. Proof goes here.
Prove that the set W = {(x, y, z) * + = 0} is a subspace of Rs and then find a basis in W.
Prove that {19a +37b| : a,b E Z} = NU {0}.
Prove that {19a +37b| : a,b E Z} = NU {0}.
problem 3
SEL 3. Prove that the mapping w 4. Prove that w z3 3z 1 is one-to-one for |z <1 Z n S {z| |z < 1} is continuous at z 1 +z6 5. Find the lim
5. For z, w E C, show the following identities. (a) z + w = z + W (b) zw = zw (c) |zw| = |2||w (d) () = 1 where w #0 (e) [2"| = |z|" where n is a positive or negative integer
detailed solution for this one ?????
11. (a) Gi) If w=z+z-' prove that (i) z2 + z 2 = w2 -2 ; 24 +2° + z²+z+1 = z2 (W2 + w+1) = (z? +[1+V5]+1)(22 +[1–V5]+1). (b) Show that the roots of 24 +2+z2+z+1=0 are the four non-real roots of z' =1. (c) Deduce that cos 72° = +(15 – 1) and cos 36° = (15+1).