1. The words in the usual dictionary are partitioned into 27 subsets (so called alphabetically).
i) State the equivalence relation on the set of words whose equivalence classes are exactly those 27 subsets.
ii) Find that relation you defined indeed is equivalence relation.
iii) Give another equivalence relation which can partition words in dictionary. Prove that it is indeed the equivalence relation. Describe equivalence classes.
2. Assume that we send eight-digit numbers (data) plus two-digit check numbers, total of ten-digit numbers over network. Find the method of using 2-digit check numbers with which receiver can check if first 8 digit numbers were correctly transmitted and received by looking at last two-digit check numbers. Method must use concept of congruence modulo m. Select number m so that probability of getting same 2-digit check number from 2 different eight-digit numbers is minimal. Illustrate all work.