Given the following countable listing of infinite lists wher

Given the following countable listing of infinite lists, where each component x_ij is either 0 or 1. x_00, x_01, ..., x_0n, ... x_10, x_11, x_12, ..., x_1n, ... x_20, x_21, x_22, ... x_2n, ... Describe another infinite list where each component is either 0 or 1 that is not in this listing.

Solution

Consider a list with the diagonal component of the given lists. ie. (x₀₀, x₁₁, x_22, ..........x_nn, ........). Now the list will be the complement of the diagonal component list. That means , if the x₀₀ is 0, then the new list contains first elemnt as 1 and  if the x₀₀ is 1, then the new list contains first elemnt as 0 and so on. So the new required list will be  (x₀₀\', x₁₁\', x22\', ..........x_nn\', ........). This is infinite list which is no in the given listing.