Prove that there is no integral domain with exactly six elements. Canyour argument be adapted to show that there is no integral domain with exactly four elements? What about 15 elements? Use these observationsto guess a general result about the number of elements in afinite integral domain.

There is a reason all our definitions in this class begin with ∀ > 0. Here are two results which can be considered two sides of the same concept – I consider it the fundamental observation of analysis – which is the reason for this ubiquity of ∀ > 0. a) Let c be a nonnegative number, i.e. c ≥ 0. Suppose c ≤  for all  > 0. Prove that c = 0.

In how many ways can we arrange all the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? (Note: IINNOOFRMTA is allowed but not INFORINMOTA, as IN occurs twice)

please don't provide handwritten solution...