Chapter 13 Mathematic Structures 13.1 Modular Arithmetic Definition 1 (modulo). Let a be an integer and m be a positive integer. We denoted by a mod m the remainder when a is divided by m. Theorem 1 is an equivalence relation. Example 1 17 mod 5 = 2, -133 mod 9=2, and 2001 mod 101 = 82.

It is convenient to use i represents. can be simplified as

13.2 Groups Definition 1 A group (G,*) consists of a set G and a binary operation * which satisfies the following conditions: (1)(closure) For all x and y in G, x*y is in G. (2)(Associativity) For all x, y, z in G, (x*y)*z = x*(y*z) (3)(Identity) There is an element e in G such that e*x = x*e (4)(Inverse) For any x in G, there is an x’ in G such that x*x’=x’*x=e Example 1 Let Z be the set of integer, and +, –, × are the integer operations on Z. (Z, +) is a group. (Z, – ) and (Z, ×) are not a groups (note: e = 0 in (Z,+) and e = 1 in (Z, ×)). Example 3 Example 2 Let R be the set of real number. (R, ×) is an group.

Example 4 Example 5 Definition 2 A group (G,*) is commutative if for any a, b in G, a*b = b*a.

13.3 Rings

13.4 Fields