Let A be an n × n matrix which is not 0 but A2 = 0.
Let I be the identity matrix.
a)Show that A is not diagonalizable.
b)Show that A is not invertible.
c)Show that I-A is invertible and find its inverse.
Starting with the expression for Pr[A1 + A2], show that for
three events Pr[A1 + A2 + A3] = Pr[A1] + Pr[A2] + Pr[A3] −
Pr[A1A2]
− Pr[A1A3] − Pr[A2A3] + Pr[A1A2A3]
Let L = {aibj | i ≠ j; i, j ≥ 0}.
Design a CFG and a PDA for this language. Provide a direct
design for both CFG and PDA (no conversions from one form to
another allowed).
Let f : [0, 1] → R and suppose that, for all finite subsets of
[0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1,
we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0,
1] : f(x) ̸= 0}. Show that S is countable