What is a bijection between R and Z x [0,1) and how do I prove it is indeed bijection? (R is all real numbers and Z represents integers)

(c) Prove that 10 is not separable.

(a) How many triples (x, y, z) of nonnegative integers satisfy x+y+z= 100? ww many triples (x, y, z) of nonnegative integers satisfy x+y+z= 100 with 0 ≤ x ≤ 2?

This is question of Complex Analysis .

Consider four statements Re(z1) Re(z2) (B) |21 · z2| = |Z1|· |2| (C) {1< |z – 1| < 2} is a connected set • (D) arg(z1 + z2) = arg(z1) + arg(z2) 21 • (A) Re 22 Which of the following is correct for all complex numbers? Select one: a. None of these O b. (A) and (D) only С. (C) only d. (B) only O e. (B) and (C) only е. Clear my choice Check

please send handwritten solution for part c

(ii) Find the rank and nullity of the [2 0 4 -7 2 4 6 A |1 -5 2 [7 -9 2 -4] 4 Verify that values obtained verify the dimension theorem.

Ans. no. 4 only.

Prove the following statement using the axioms and lemmas For all nonzero real numbers w, x and y, (w · x) · multinv(w · y) = x · multinv y.

g) Prove that f : R → R where f(x) = x² – x is not injective. h) Disprove the following claim by counter example. "For all natural numbers x, x² is odd". Show your work