1. 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

2. 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

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

4. 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

5. 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

6. 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

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

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

9. 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

10. 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

11. 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

12. 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

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

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

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

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

17. 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

18. 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
3 

19. 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
3  9 9

20. 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.