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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 a uniquely determined element a b = c where c S

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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?

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Power Set Operation Is a binary operation on (A)? Is a binary operation on (B)?

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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?

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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)

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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

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

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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

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Discussion Let (A) be the power set of A. Which binary operation properties hold for ? For ?

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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)

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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.

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Finite Sets and Operations Power Set of a Finite Set Rigid Motions of a Figure

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

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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

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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

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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

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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

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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

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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

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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.

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

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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.

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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.

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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.

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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/Cayl ey.html http://www-groups.dcs.st- and.ac.uk/~history/Mathematicians/Cayl ey.html

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

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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.

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Equilateral Triangle Symmetry 1 32

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Rotation 1( 1 ) 1 2 3 1 32

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Rotation 2( 2 ) 1 2 3 3 1 2

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Rotation 3( 3 ) 1 2 3 2 31

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Reflection 1(r 1 ) 1 2 2 3 1 3

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Reflection 2(r 2 ) 1 2 3 3 2 1

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Reflection 3(r 3 ) 1 2 3 1 2 3

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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?

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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.

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

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

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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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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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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

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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

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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)

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

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

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Modular Arithmetic Cayley Table for Mod 4 +

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http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm Thank You..!! http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm http://mandala.co.uk/permutations/ http://akbar.marlboro.edu/~mahoney/courses/Spr00/rubik.html

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