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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(AB) If B A, then a-b = n(A\B)

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**For any sets A and B, a b = n(AB)**

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 ab = 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 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 .

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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 2 3

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Rotation 1(1) 1 1 2 3 2 3

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Rotation 2(2) 1 3 2 2 3 1

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Rotation 3(3) 1 2 3 1 3 2

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Reflection 1(r1) 1 1 2 3 3 2

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Reflection 2(r2) 1 3 2 3 2 1

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Reflection 3(r3) 1 2 3 2 3 1

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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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**Cayley Table for Triangle Symmetries**

r r r3 1 2 3 r1 r2 r3

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**What is the identity symmetry?**

Is closed? Is commutative?

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**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?

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**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?

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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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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.

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**Is the dihedral group commutative?**

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?

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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/modart/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**

Thank You..!!

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Discrete Mathematics II

Discrete Mathematics II

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