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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. Discussion. Can you determine some other binary operations on the whole numbers?

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binary operations

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

discussion
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?
power set operation
Power Set Operation
  • Is  a binary operation on (A)?
  • Is a binary operation on (B)?
whole number subsets
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?

binary operation properties
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)
slide7
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

discussion8
Discussion
  • Which of the binary operation properties hold for multiplication over the whole numbers?
  • What about for subtraction over the integers?
exploration
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
discussion10
Discussion

Let (A) be the power set of A.

  • Which binary operation properties hold for  ?
  • For  ?
set definitions of operations
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)
slide12
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.

finite sets and operations
Finite Sets and Operations
  • Power Set of a Finite Set
  • Rigid Motions of a Figure
exploration14
Exploration

Let A = {a,b}, then (A) has 4 elements:

S1 =ø

S2 = {a}

S3 = {b}

S4 = {a,b}

slide15
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

slide16
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

slide17
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

slide18
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

slide19
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

slide20
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

exploration extension
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.

arthur cayley

Arthur Cayley

Born: 16 Aug 1821 Died: 26 Jan 1895

slide23
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.
slide24
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.
slide25
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.
slide26
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
permutation of a set
Permutation Of A Set

Let S be a set.

A permutation of the set S is a 1-1 mapping of S onto itself.

symmetry of geometric figures
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.

composition operation
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?

exploration37
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.

slide38

Cayley Table for Triangle Symmetries

123 r1 r2 r3

1

2

3

r1

r2

r3

slide39
What is the identity symmetry?
  • Is  closed?
  • Is  commutative?
slide40

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?

slide42

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?

groups

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

slide45

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.

slide46

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?
modern art
Modern Art

Cayley Table and Modular Arithmetic Art

Website:http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm

slide49

Modular Arithmetic

Cayley Table for Mod 4 +

slide52

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

http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm

http://ccins.camosun.bc.ca/~jbritton/modart/jbmodart2.htm

http://mandala.co.uk/permutations/

http://akbar.marlboro.edu/~mahoney/courses/Spr00/rubik.html

Thank You..!!