4.5 Finite Mathematical Systems We will be looking at operations on finite sets of numbers. We will denote the operation by a generic symbol, like *. What is an operation? An operation is a rule for combining two elements of a set. Example 1 The set {a, b, c, d} and operation * where a a = a b a = b c a = c d a = d a b = b b b = d c b = a d b = c a c = c b c = a c c = d d c = b a d = d b d = c c d = b d d = a An easier way to display an operation is by an Operation Table. The previous example could have been put in the following table. * a b c d a a b c d b b d a c c c a d b d d c b a The important properties to look for in a system are the following: closure, commutative, associative, identity and inverse. Properties of a single operator system: Closure: The system is closed if for all elements a and b, is a set of the system. Commutative: The system has the commutative property if for all elements a and b, a b a b = b a. We can test for commutative if the table passes the diagonal line test. If the table is symmetric about the main diagonal then it is commutative. Associative: The system has the associative property if for all elements a, b and c, (a b) c = a (b c). To test to see if the system has the associative property you must test at least TWO examples. 1
Identity: The system has an identity element e (where e is in the set) if for all elements a. e a = a e = a Inverse: The system has the inverse property if for every element a of the system, there is an element x of the system such that a x = e = x a where e is the identity element. Example 2: Decide which of the properties of single operation systems are satisfied. If the identity property is satisfied, give the identity element. If the inverse property is satisfied, give the inverse of each element. If the identity property is satisfied, but the inverse property is not, name the elements that have no inverses. 1. {1, 2, 3, 4}; operation 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 2
2. {1, 2, 3, 4, 5}; operation 1 2 3 4 5 1 1 2 3 4 5 2 2 4 0 2 4 3 3 0 3 0 3 4 4 2 0 4 2 5 5 4 3 2 1 3. {r, s, t, u}; operationz Z r s t u r u t r s s t u s r t r s t u u s r u t 3
4.6 Groups Group: A group is a mathematical system (set of elements and an operation) that satisfies the closure, associative, identity and inverse properties. Example 1: Does the set -1, 1, under the operation of multiplication form a group? What about under addition? Example 2: Does the set of integers {..., 3, 2, 1, 0, 1, 2, 3,...} under the operation of addition form a group? What about multiplication? 4
4.4 Clock Arithmetic and Modular Systems A mathematical system has 3 major properies. 1. It is a set of elements 2. It has one or more operations to combine these elements (ie. Multiplication, addition) 3. It has one or more relations for comparing those elements. We will be looking a the 12-hour clock system which is based on an ordinary clock face, except that 12 is considered to be the 0 hour and we don t consider minutes, only whole number divisions. Thesetweuseis {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. We add numbers the same way we would add times on a clock. For example if we start at 5 and add 2 hours to it we end up at 7. If we start with 5 and add 7 hours to it we end up at 12 but in clock arithmetic that is considered to be 0. If we start at 5 and add 8 hours to it we get one hour past zero so we end up at 1. So in clock arithmetic. 5+2=7, 5+7=0, and 5 + 8 = 1 Example 1: Perform the following operations using 12 hour clock arithmetic. 8+8= 2+8= 0+3= 4+9= Properties of a single operator system: Closure: The system is closed if for all elements a and b, a + b is a set of the system. Commutative: The system has the commutative property if for all elements a and b, a + b = b + a. Associative: The system has the associative property if for all elements a, b and c, (a + b)+c = a +(b + c). 5
Identity: The system has an identity element e (where e is in the set) if e + a = a + e = a for all elements a. Does clock arithmetic satisfy the following properties? closure property? commutative property? associative property? identity property? Since we can do addition on a clock we can also do subtraction. Find the following. 8 3= 2 8= 0 3= 4 9= We can also do multiplication: 5 5=5+5+5+5+5=10+10+5=8+5=1 Example 2: Find each product using 12 hour clock arithmetic. 8 3= 2 8= 0 3= 4 9= 6
Additive Inverse When we talk about the additive inverse we want to know what number, x, towehaveto add to a so that a + x = 0? In regular addition this is easy but in clock addition it is a little more difficult. Since 0 is the same as 12 in 12-hour clock arithmetic then the additive inverse is whatever number you need to add to a toget12. Forexample5+7=0so7istheadditive inverse of 5 in 12-hour clock arithmetic. Fill in the following table: Clock value a Additive inverse of a 0 1 2 3 4 5 7 6 7 8 9 10 11 Modular Systems Congruence Modulo m The integers a and b are congruent modulo m (where m is a natural number greater than 1 called the modulus if and only if the difference a - b is divisible by m. Symbolically, this congruence is written a b(modm). Clock arithmetic is a system of congruences modulo 12. 7