Binary Operations

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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 ab-1
• a ? ba 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/Mathem
  aticians/Cayley.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
3
2
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Rotation 1(??1)
1
1
2
3
3
2
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Rotation 2(??2)
1
3
2
2
3
1
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Rotation 3(??3)
1
2
3
3
1
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
1
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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Cayley Table for Triangle Symmetries

• ?1 ?2 ?3 r1
  r2 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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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
• Websitehttp//ccins.camosun.bc.ca/jbritton/modar
  t/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/Mathem
aticians/Cayley.html
http//ccins.camosun.bc.ca/jbritton/modart/jbmoda
rt2.htm
http//ccins.camosun.bc.ca/jbritton/modart/jbmoda
rt2.htm
http//mandala.co.uk/permutations/
http//akbar.marlboro.edu/mahoney/courses/Spr00/r
ubik.html

Thank You..!!