Chapter 0 (Preliminaries)
Q.) How would you prove the converse?
A partition of a set S defines on equivalence relation on S.
Hint: Define a relation as X - Y if X and Y are elements of the same subset of the partition.
10.) Let n be fixed positive integer greater than 1. If a mod n = a' and b mod n = b' .Prove that (a+b) mod n = (a' + b') mod n and (ab) mod n = (a'b') mod n.
11.) Let "n" be a fixed positive integer greater than 1. If a mod n = a' and b mod n = b'. Prove that (a+b) mod n = (a'+b') mod n and
(ab) mod n = (a'b') mod n.
30.) Prove that for every integer n, n3 mod 6 = n mod 6.
41.) The International Standard Book Numbers (ISBN) a1, a2,a3, a4, a5 , a6, a7,a8,a9,a10 has the property (a1, a2,....................a10) . (10, 9,8, 7, 6, 5, 4, 3, 2, 1) mod 11 = 0. The digits a10 is the check digit. When a10 is required to be 10 to make the dot product 0, the character X is used as the check digit. Verify the check digit for the ISBN assigned to this book.
49.) Let S be the set of integers. If a, b ε S, define aRb if ab > = 0. If R an equivalence relation on S?
50.) Let S be the set of integers. If a, b ε S, define aRb if a + b is even. Prove that R is an equivalence relation and determine the equivalence classes of S.
This is a set of abstract algebra questions involving proofs of modular arithmetic and equivalence classes.