Question 1 1.Prove: If z,, zz, ..., Zn are complex numbers, then z1 + z2 + ** + zn = Z + Z,+ ...+ z(wltu), and b =(wl-u. Prove z2 = w if v 2 02.Suppose = a + bi , w = u + vi, a =and that (z)² = w if v <0. Conclude that every complex number except for 0 has atleast two square roots3. Let z be a complex number such that |z| = 1, that is z Z = 1. Compute |1+ zl? +|1 – z|2. (4.a. If k 22, for each XE RK, prove there exist a ỹ E R such that y # Ō, but z -ỹ = 0b. For each e R , does there exist a y E R such that y # 0, but x y = 05. Prove that | i +ỹ|2 + |x - y |2 = 2|x|2 + 2|y|² if xE R and y e R*6. Let f be a function that maps Xinto Y (f:X→ Y ). Let | a E A} be an indexed family ofsubsets of Y. Show f-1(naeA Ga) = NaeAf(Ga)7. Prove T = { ..., –56, -42, -28,-14,0, 12, 28, 42, ... } is a countable set8. Let J = {1,2, 3, ... }, the set of counting numbers and let R be the set of real numbers, IsR - / countable? ExplainPag11.759.75.75

Name: Question 1 1.Prove: If z,, zz, ..., Zn are complex numbers, then z1 +
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Description: Question 1 1.Prove: If z,, zz, ..., Zn are complex numbers, then z1 + z2 + ** + zn = Z + Z,+ ...+ z(wltu), and b =(wl-u. Prove z2 = w if v 2 02.Suppose = a + bi , w = u + vi, a =and that (z)² = w if v <0. Conclude that every complex number except for