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
Power Set Operation
¾Can you determine some other binary
operations on the whole numbers?
¾Is ∩ a binary operation on ℘(A)?
¾Can you make up a “binary operation”
over the integers that fails to satisfy the
uniqueness criteria?
¾Is ∪ a binary operation on ℘(B)?
Whole Number Subsets
Binary Operation Properties
¾Let E = set of even whole numbers.
Are + and • binary operations on E?
Let ° be a binary operation defined on
the set A.
¾Let O = set of odd whole numbers.
Are + and • binary operations on O?
¾Closure Property: For all x,y ∈ A
x°y∈A
¾Commutative Property: For all x,y ∈ A
x ° y = y ° x (order)
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Discussion
¾Associative Property: For all x,y,z ∈A
x ° ( y ° z )=( x ° y ) ° z
¾Identity: e is called the identity for the
operation if for all x ∈ A
x°e=e°x=x
¾Which of the binary operation properties
hold for multiplication over the whole
numbers?
¾What about for subtraction over the
integers?
Exploration
Define a binary operation ° over the
integers. Determine which properties of
the binary operation hold.
Discussion
Let ℘(A) be the power set of A.
¾Which binary operation properties hold for
∩?
¾a°b=b
¾ a ° b =larger of a and b
¾ a ° b = a+b-1
¾For ∪ ?
¾ a ° b=a+ b+ ab
Set Definitions of Operations
Let a, b ∈ Whole Numbers
Let A, B be sets with n(A) = a and
n(B)=b
¾ If A ∩ B = ø (Disjoint sets),
then a + b = n(A∪B)
¾For any sets A and B, a • b = n(A×B)
¾For any set A and whole number
m,
a ÷ m = partition of n(A) elements of A
into m groups.
¾If B ⊆ A, then a-b = n(A\B)
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Finite Sets and Operations
• Power Set of a Finite Set
• Rigid Motions of a Figure
• 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
• 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
Exploration
Let A = {a,b}, then ℘(A) has 4 elements:
S1 = ø
S2 = {a}
S3 = {b}
S4 = {a,b}
• 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
• 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
Exploration Extension
• 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
Arthur Cayley
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.
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.
¾The most important of his work was
developing the algebra of matrices,
work in non-Euclidean geometry and ndimensional geometry.
¾As early as 1849 Cayley wrote a paper
linking his ideas on permutations with
Cauchy's.
¾ In 1854 Cayley wrote two papers which
are remarkable for the insight they have
of abstract groups.
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¾At 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.
¾These tables become known as Cayley
Tables.
Permutation Of A Set
Let S be a set.
A permutation of the set S is a 1-1 mapping
of S onto itself.
¾He 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 .
¾http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Cayl
ey.html
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.
Rotation 1( ρ1)
Equilateral Triangle Symmetry
1
1
2
3
2
1
3
2
3
5
Rotation 2( ρ2)
Rotation 3( ρ3)
1
1
2
3
2
3
2
3
1
Reflection 1(r1)
1
1
2
2
3
Reflection 3(r3)
1
3
Reflection 2(r2)
1
3
1
2
3
3
2
1
2
Composition Operation
The operation for symmetry a ° b is the
composition of symmetry a followed by
symmetry b.
2
Example:
1 2 3 1 2 3 1 2 3
o
=
2 3 1 1 3 2 3 2 1
ρ 2 o r1 =
3
2
3
1
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.
Cayley Table for
Triangle Symmetries
r1
r2
r3
r1
r2
r3
r1
r2
r3
r1
r2
r3
Exploration Extension
• What is the identity symmetry?
Q1: Find the symmetries of a square.
How many elements are in this set?
• Is ° closed?
• Is ° commutative?
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?
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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
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.
Here is a colorful table for the dihedral
group of order 5
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
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 +
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..!!
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