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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?
Can you make up a “binary operation”
over the integers that fails to satisfy the
uniqueness criteria?
Power Set Operation
Is  a binary operation on (A)?
Is a binary operation on (B)?
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
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)
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
Discussion
Which of the binary operation properties
hold for multiplication over the whole
numbers?
What about for subtraction over the
integers?
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
Discussion
Let (A) be the power set of A.
Which binary operation properties hold for
 ?
For  ?
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)
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
• Power Set of a Finite Set
• Rigid Motions of a Figure
Exploration
Let A = {a,b}, then (A) has 4 elements:
S1 = ø
S2 = {a}
S3 = {b}
S4 = {a,b}
• 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
• 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
• 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
• 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
• 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
• 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
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
Born: 16 Aug 1821 Died: 26 Jan 1895
• 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.
The most important of his work was
developing the algebra of matrices,
work in non-Euclidean geometry and ndimensional 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.
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.
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.stand.ac.uk/~history/Mathematicians/Cayl
ey.html
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
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.
Equilateral Triangle Symmetry
1
2
3
Rotation 1(1)
1
2
1
3
2
3
Rotation 2(2)
1
3
3
2
2
1
Rotation 3(3)
1
3
2
2
1
3
Reflection 1(r1)
1
3
1
2
2
3
Reflection 2(r2)
1
3
3
2
1
2
Reflection 3(r3)
1
3
2
2
3
1
Composition Operation
The operation for symmetry a  b is the
composition of symmetry a followed by
symmetry b.
Example:
 1 2 3  1 2 3   1 2 3 
2  r1  
  
  

 2 3 1  1 3 2   3 2 1 
What is the resulting symmetry from this product?
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.
Cayley Table for
Triangle Symmetries
1
r3
1
2
3
r1
r2
r3
2
3
r1
r2
• What is the identity symmetry?
• Is  closed?
• Is  commutative?
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?
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
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
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.
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?
Here is a colorful table for the dihedral
group of order 5
Modern Art
Cayley Table and Modular Arithmetic Art
Website:http://ccins.camosun.bc.ca/~jbritton/mo
dart/jbmodart2.htm
Modular Arithmetic
Cayley Table for Mod 4 +
http://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..!!