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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.

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Presentation on theme: "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."— Presentation transcript:

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

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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.  and.ac.uk/~history/Mathematicians/Cayl ey.html and.ac.uk/~history/Mathematicians/Cayl ey.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 32

30 Rotation 1(  1 )

31 Rotation 2(  2 )

32 Rotation 3(  3 )

33 Reflection 1(r 1 )

34 Reflection 2(r 2 )

35 Reflection 3(r 3 )

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  1  2  3 r 1 r 2 r 3  1  2  3 r 1 r 2 r 3 Cayley Table for Triangle Symmetries

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

40 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? Exploration Extension

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42 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? Exploration Extension

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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 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. Dihedral Groups

46 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? Non-Abelian Group (non-commutative)

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/mo dart/jbmodart2.htm

49 Modular Arithmetic Cayley Table for Mod 4 +

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52 Thank You..!!


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