A mathematical system - that is an algebra - consists of at least one set with at least one operation - for example: the set of integers with the operation addition. Algebraic Structure

Example 1: set = { a, b, c, d } operation is * as defined by the table below: a*c = b b b*b = d d c d c*a = d d a c a a c*b = a a d f This operation has been defined by randomly completing the table. While it is an example of an algebra, it is NOT ORGANIZED! c b b a

Compare: Example 1: set = { a, b, c, d } operation is * b d a c f set = { p, q, r, s } operation is # ORGANIZED DISORGANIZED

Compare: c*d is NOT a member of the set { a, b, c, d } Every entry is a member of the set { p, q, r, s } Does NOT have CLOSURE Has CLOSURE b d a c f DISORGANIZED ORGANIZED

Compare: for ALL x and y: x#y = y#x p#q = q#p b*a  a*b q#s = s#q Does NOT have COMMUTATIVITY Has COMMUTATIVITY Note the symmetry. b d a c f DISORGANIZED ORGANIZED

Compare: for ALL x and y: x#y = y#x b*a  a*b Does NOT have COMMUTATIVITY Has COMMUTATIVITY Note the symmetry. b d a c f DISORGANIZED ORGANIZED

Compare: ( c*b )* d c*( b*d ) a * d c* c b d a c f DISORGANIZED

Compare: ( c*b )* d  c*( b*d ) a * d c* c a d b d a c f DISORGANIZED

Compare: ( c*b )* d  c*( b*d ) ( p # s )# q = p#( s # q ) a * d c* c r # q p# p a d b d a c f DISORGANIZED ORGANIZED

Compare: ( c*b )* d  c*( b*d ) ( p # s )# q = p#( s # q ) a * d c* c r # q p# p a d q q b d a c f DISORGANIZED ORGANIZED

Compare: ( c*b )* d  c*( b*d ) ( x # y )# z = x #( y # z ) for ALL x, y, and z Does NOT have ASSOCIATIVITY Has ASSOCIATIVITY b d a c f DISORGANIZED ORGANIZED

Compare: r is the IDENTITY Has NO IDENTITY r#x = x for ALL x r#p=p r#q=q r#r=r r#s=s b d a c f DISORGANIZED ORGANIZED

Compare: r is the IDENTITY x#r = x for ALL x b d a c f DISORGANIZED

Compare: r is the IDENTITY p is the INVERSE of s p#s = s#p = r b d a c DISORGANIZED ORGANIZED

Compare: r is the IDENTITY q is the INVERSE of q q#q = r b d a c f DISORGANIZED ORGANIZED

example 3: set = { 1, 3, 7, 9 } operation =  a b= the units digit of ab 7  9 = 6 3 7  9 = 3

example 3: set = { 1, 3, 7, 9 } operation =  a b= the units digit of ab 7 9 = 6 3 closure commutativity identity inverses 3 is the inverse of 7 9 is its own inverse

example 3: set = { 1, 3, 7, 9 } operation =  a b= the units digit of ab 7 9 = 6 3 associativity (3  9)  7 3  (9  7) 7  7 3  3

example 3: set = { 1, 3, 7, 9 } operation =  a b= the units digit of ab 7 9 = 6 3 associativity (3  9)  7 3  (9  7) 7  7 3  3 9 9

definition: A mathematical system that has the following four properties is called a GROUP: closure associativity identity every element has an inverse example 3 (slide 16) is an example of a group.