Binary Operations

1. Binary Operations Let S be any given set. A binary operation  on S is a correspondence that associates with each ordered pair (a, b) of elements of S a uniquely determined element a  b = c where c  S

3. Discussion • Can you determine some other binary operations on the whole numbers? • Can you make up a “binary operation” over the integers that fails to satisfy the uniqueness criteria?

4. Power Set Operation • Is  a binary operation on (A)? • Is a binary operation on (B)?

5. Whole Number Subsets • Let E = set of even whole numbers. Are + and  binary operations on E? • Let O = set of odd whole numbers. Are + and  binary operations on O?

6. Binary Operation Properties Let  be a binary operation defined on the set A. • Closure Property: For all x,y  A x  y  A • Commutative Property: For all x,y  A x  y = y  x (order)

7. Associative Property: For all x,y,z A x  ( y  z )=( x  y )  z • Identity: e is called the identity for the operation if for all x  A x  e = e  x = x

8. Discussion • Which of the binary operation properties hold for multiplication over the whole numbers? • What about for subtraction over the integers?

9. Exploration Define a binary operation  over the integers. Determine which properties of the binary operation hold. • a  b = b • a  b =larger of a and b • a  b = a+b-1 • a  b=a+ b+ ab

10. Discussion Let (A) be the power set of A. • Which binary operation properties hold for  ? • For  ?

11. Set Definitions of Operations Let a, b  Whole Numbers Let A, B be sets with n(A) = a and n(B)=b • If A  B =ø (Disjoint sets), then a + b = n(AB) • If B A, then a-b = n(A\B)

12. For any sets A and B, a  b = n(AB) • For any set A and whole number m, a m = partition of n(A) elements of A into m groups.

13. Finite Sets and Operations • Power Set of a Finite Set • Rigid Motions of a Figure

14. Exploration Let A = {a,b}, then (A) has 4 elements: S1 =ø S2 = {a} S3 = {b} S4 = {a,b}

15. Define + on the Power Set by a table + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

16. Is + a binary operation? Is it closed? + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

17. Does an identity exists? If so, what is it? + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

18. Is the operation commutative? How can you tell from the table? + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

19. Can the table be used to determine if the operation is associative? How? + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

20. Determine a definition for the operation + using ,  and \ + S1 S2 S3 S4 S1 S1 S2 S3 S4 S2 S2 S1 S4 S3 S3 S3 S4 S1 S2 S4 S4 S3 S2 S1

21. Exploration Extension Suppose for (A) that ab = a  b. Q1: Construct an operation table using this definition. Q2: What is the identity for a  b? Q3: Does the distributive property hold for a(b + c) = (a  b) +(a  c)? Try a few cases.

22. Arthur Cayley Born: 16 Aug 1821 Died: 26 Jan 1895

23. In 1863 Cayley was appointed Sadleirian professor of Pure Mathematics at Cambridge. • He published over 900 papers and notes covering nearly every aspect of modern mathematics.

24. The most important of his work was developing the algebra of matrices, work in non-Euclidean geometry and n-dimensional geometry. • As early as 1849 Cayley wrote a paper linking his ideas on permutations with Cauchy's. • In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups.

25. At that time the only known groups were permutation groups and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. • These tables become known as Cayley Tables.

26. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices were groups . • http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html

27. Permutation Of A Set Let S be a set. A permutation of the set S is a 1-1 mapping of S onto itself.

28. Symmetry Of Geometric Figures A permutation of a set S with a finite number of elements is called a symmetry. This name comes from the relationship between these permutations and the symmetry of geometric figures.

29. Equilateral Triangle Symmetry 1 2 3

30. Rotation 1(1) 1 1 2 3 2 3

31. Rotation 2(2) 1 3 2 2 3 1

32. Rotation 3(3) 1 2 3 1 3 2

33. Reflection 1(r1) 1 1 2 3 3 2

34. Reflection 2(r2) 1 3 2 3 2 1

35. Reflection 3(r3) 1 2 3 2 3 1

36. Composition Operation The operation for symmetry a  b is the composition of symmetry a followed by symmetry b. Example: What is the resulting symmetry from this product?

37. Exploration Complete the Cayley Table for the symmetries of an equilateral triangle. To visualize the symmetries form a triangle from a piece of paper and number the vertices 1, 2, and 3. Now use this triangle to physically replicate the symmetries.

38. Cayley Table for Triangle Symmetries 123 r1 r2 r3 1 2 3 r1 r2 r3

39. What is the identity symmetry? • Is  closed? • Is  commutative?

40. Exploration Extension Q1: Find the symmetries of a square. How many elements are in this set? Q2: Make a Cayley Table for the square symmetries. What operation properties are satisfied?

42. Exploration Extension Q3: How many elements would the set of symmetries on a regular pentagon have? A regular hexagon? Q4: Try this with a rectangle. How many elements are in the set of symmetries for a rectangle?

44. Groups A nonempty set G on which there is defined a binary operation ° with Closure: a,b  G, then a ° b  G Identity:  e  G such that a ° e = e ° a = a for  a  G Inverse: If a  G, x  G such that a ° x = x ° a = e Associative: If a, b, c  G, then a ° (b ° c) = (a ° b) ° c

45. Dihedral Groups One of the simplest families of groups are the dihedral groups. These are the groups that involve both rotating a polygon with distinct corners (and thus, they have the cyclic group of addition modulo n, where n is the number of corners, as a subgroup) and flipping it over.

46. Non-Abelian Group (non-commutative) • Is the dihedral group commutative? • Since flipping the polygon over makes its previous rotations have the effect of a subsequent rotation in the opposite direction, this group is not commutative. • Is the dihedral group the same as the permutation group?

47. Here is a colorful table for the dihedral group of order 5

48. Modern Art Cayley Table and Modular Arithmetic Art Website:http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm

49. Modular Arithmetic Cayley Table for Mod 4 +