1. Cardinality and Algebraic Structures
Dr Tijl De Bie
Dept. Eng. Maths.

2. Contents Part I (weeks 1-7) 1 Introduction
2 Combinatorics, permutations and combinations.
3 Algebraic Structures and matrices: Homomorphism, isomorphism, group, semigroup, monoid, rings, fields
4 Lattices and Boolean algebras
If time remains: some illustrations of the use of group theory in cryptography
Part II (weeks 8-12)
Vector spaces

3. Introduction Computer programs frequently handle real world data.
This data might be financial e.g. processing the accounts of a company.
It may be engineering data e.g. from sensors or actuators in a robotic system.
It may be scientific data e.g. weather data or geological data concerning rock strata.
In all these cases data typically consists of a set of discrete elements.
Furthermore there may exist orderings or relationships among elements or objects.
It may be meaningful to combine objects in some way using operators.
We hope to clarify our concepts of orderings and relationships among elements or objects
We look at the idea of formal structures such as groups, rings and and formal systems such as lattices and Boolean algebras

4. Number Systems
The set of natural numbers is the infinite set of the positive integers. It is denoted N and can have different representations:
{1,2,3,4, }
{1,10,11,100,101,.....}
are alternative representations of the same set expressed in different bases. Nm is the set of the first m positive numbers i.e. {1,2,3,4, ,m}. N0 is the set of natural numbers including 0 i.e. {0,1,2,3,5,....}
Q denotes the set of rational numbers i.e. signed integers and fractions
{0,1,-1,2,-2,3,-3,....,1/2,-1/2,3/2,-3/2,5/2,
-5/2,....,1/3,-1/3,2/3,-2/3, }
R is the set of real numbers i.e. the coordinates of all the points on a line.
Z is the set of all integers, both positive and negative {0,1,-1,2,-2,3,-3,......}

5. 2 Combinatorics: Permutations
A permutation of the elements of a set A is a bijection from A onto itself.
If A is finite we can calculate the number of different permutations. Suppose A={a1,...,an} n
choices
n-1
choices 1
choice a1 a2 an
total number of ways of filling the n boxes
n x (n-1)x(n-2)x(n-3) x1=n!
nPn=n!
eg a possible permutation of {1,2,3,4,5,6} is

6. Composition of Permutations
If :A A and :A A are permutations of A then the composition or product .of  and satisfies for all x in A
.x)= (x))
Notice that since both and are bijections from A into A so is . In other words . is a permutation of A.
Example: Let A={1,2,3,4,5,6} then two possible permutations are
For . we have that

7. Cyclic Permutations A cyclic permutation on a set A of n elements has
the form where :
For shorthand we often write
 is said to be a k cycle
Example
or (6 1 4) is a cyclic
permutation
Two cyclic permutations
and
are said to be disjoint if
e.g. (4 5 2) and (3 1 6) are disjoint

8. Notice that
Other examples are or
Can you spot a product of disjoint cyclic
permutations equivalent to the following
permutation ?

9. Theorem: Every permutation of a finite set A can be expressed as a combination of disjoint cycles.
Structure underlying permutations
Note that the following hold:
(1) The product of two permutations is a uniquely determined permutation of the same set.
(2) The composition of permutations is associative.
(3) The permutation
is called the identity permutation and has the
property that
(4) For every permutation
there is an inverse
such that

10. Combinations
When we think about combinations we do not allow repeats and unlike permutations we do not consider order.
Combinations look at the number of different ways of picking a subset of k elements from a set of n elements.
Think of the number of ways of picking a list of k distinct elements of n
no. of choices n
n-1
n-k-2
n-k-1
places
= n(n-1)(n-2) (n-k-1) = n!/(n-k)!
For each possible list there are k! permutations
so since we are not interested in order we
should divide the above by k!.
C(n,k) = Cnk = n!/(n-k)!k!

11. Example: Choosing 2 elements from {a,b,c,d}
{a,b},{a,c},{a,d},
{b,c},{b d},{c,d}
C(4,2)= 4!/(2! 2!) =6
Combinations with Repetitions
We could also consider combinations with repetitions. With repetitions the number of distinct combinations of k elements chosen from n is:
C(n+k-1,k)= (n+k-1)!/k!(n-1)!
Number of different throws of 2 identical dice
(1 1)(2 2)(3 3)(4 4)(5 5)(6 6)
(1 2)(1 3)(1 4)(1 5)(1 6)
(2 3)(2 4)(2 5)(2 6)
(3 4)(3 5)(3 6)(4 5)(4 6)(5 6)
C(7,2)=21

12. Algebraic Structures
When we consider the behaviour of permutations under the composition operation we noticed certain underlying structures.
Permutations are closed under this operation, they exhibit associativity, an identity element exists and an inverse exists for each permutation
These properties define a general type of algebraic structure called a group.
In this section we shall look at groups in more detail as well as other similar algebraic structures such as semigroups and monoids.
Later we will progress to consider more complex algebraic structures such as rings, integral domains and fields.
We will see that many real life situations are examples of these algebraic structures

13. Groups A group or is a set G with binary operation
which satisfies the following properties 1.
is a closed operation i.e. if
and
then 2.
this is the
associative law
3. G has an element e, called the identity, such that 4.
there corresponds an element
such that
Example:
The set of all permutations of a set A
onto itself is group (called the symmetric group Sn
for n elements).

14. Group of Symmetries of a Triangle
Consider the triangle X O Y Z n m
We can perform the following transformations
on the triangle
1=identity mapping from the plane to itself
p=rotation anticlockwise about O through 120
degrees
q=rotation clockwise about O through 120 degrees
a=reflection in l
b=reflection in m
c=reflection in n

15. Let
denote transformation y followed by
transformation x for x and y in {1,p,q,a,b,c}
So for example l l l Y X X a p O O O X Z m n
m Y Z n m Z Y n
Notice the table is not symmetric

16. Other examples of a group
The set of all permutations onto itself is a group (called the symmetric group Sn)
The sets of all invertible nxn matrices forms a group under ordinary matrix multiplication (called GL(n), the general linear group)
The quaternion group: Let G={I,-I,J,-J,K,-K,L,-L} where
I=[ ], J=[ ], K=[ ] , L=[ ]
j j
0 j j 0

17. Order of a group A finite group is a group where G is finite
The order of a finite group is |G|
For example if G is the set of permutations of a set A with n elements then the order of G is n!

18. Abelian Groups If is a group and is also commutative then
is referred to as an Abelian group
(the name is taken from the 19’th century
mathematician N.H. Abel)
is commutative
means that
Examples:
and
are
abelian groups.
Why is
not a group at all?

19. Modular arithmetic Recall a=b mod p iff p|a-b
Notice a=b mod p iff a=kp+b for some integer k
 a=b mod p implies p|a-b implies a-b=kp implies a=kp+b
 a=kp+b implies a-b=kp implies p|a-b implies a=b mod p

20. Modular addition
Modular addition mod 6: + 1 2 3 4 5

21. Modular multiplication
Modular multiplication mod 7: x 1 2 3 4 5 6

22. Modular multiplication
Modular multiplication mod 6: x 1 2 3 4 5

23. Modular multiplication
Not a group! (Why not?)
Which subset of {1,2,3,4,5} does form a group? x 1 2 3 4 5

24. Modular multiplication
Theorem: If n>=2 and n|p then n has no inverse under multiplication mod p
Prove it!
The subset of {1,…,p-1} relatively prime to p is a group under multiplication mod p denoted Zp*
We will clarify this on the next slides…

25. Modular arithmetic
Recall Euclid’s algorithm to find the gcd of x and y:
x=k1y+r1
y=k2r1+r2
r1=k3r2+r3 …
rn-2 =kn-1rn-1+rn
rn-1=knrn
From this… Theorem:
There exist integer a and b such ax+by=gcd(x,y)
The old remainder is divided by the new one repeatedly until the remainder is 0
The gcd is the last non zero remainder

26. Modular arithmetic
An element n has an inverse n-1 under multiplication mod p for which
n. n-1 =1 mod p
if and only if (iff) n is relatively prime to p.
Prove this!
Clearly then if p is prime then every element will have an inverse.

27. Groups in logic Consider exclusive or defined by
A⊕ B ≡(¬A∧ B)∨ (A∧ ¬B)
{t,f} is an abelian group under exclusive or.
What is the identity?
What is the inverse of t (and f)?

28. To show that an algebraic system is a group we
must show that it satisfies all the axioms of a group.
Question: Let
be a Boolean algebra
so that A is a set of propositional elements, is
like ‘or’,
is like ‘and’ and
is like ‘not’. Show
that
is an abelian group where
Answer:
(1) Associative since
prove this ?
(2) Has an identity element 0 (false) since
(3) Each element is its own inverse
(4) The operation commutes
prove this ?

29. Iterated operations a=a1 a◦a=a1 a◦a◦a=a2 a◦a…◦a=ak
(Why is this unambiguously defined?)

30. Cyclic groups
A group G is cyclic if there exists a∈G such that for any b∈G there is an integer k≥0 such that ak=b.
I.e. Every element of G is some power of a.
Element a is called the generator of G denoted G=<a>
Example:
<{1,-1},×>=<-1> since –12=1, -13=-1

31. Order of a cyclic permutation group
Show that the order is equal to p
[Show by making a drawing…]

32. Weaker structures
An Abelian group is a strengthening of the notion of group (i.e. requires more axioms to be satisfied)
We might also look at those algebraic structures corresponding to a weakening of the group axioms
Semigroup ⊆ monoid ⊆ group ⊆ Abelian Group

33. Semigroup is a semigroup if the following conditions are satisfied: 1.
is a closed operation i.e. if
and
then 2.
is associative
Example: The set of positive even integers
{2,4,6,.....} under the operation of ordinary addition
since
The sum of two even numbers is an even number
+ is associative
The reals or integers are not semigroups under -
why?

34. Monoid is a monoid if the following conditions are satisfied: 1.
is a closed operation i.e. if
and
then 2.
is associative
3. There is an identity element
Examples: Let A be a finite set of heights. Let be
a binary operation such that
is equal to
the taller of a and b. Then
is a monoid where the identity is the shortest
person in A
is a monoid:
is associative,
true is the identity, but false has no inverse
is a monoid:
is associative
false is the identity, but true has no inverse

35. Properties of Algebraic Structures
Theorem: (unique identity) Suppose that
is a monoid then the identity element is unique
Proof: Suppose there exist two identity elements
e and f. [We shall prove that e=f]
Theorem: (unique inverse) Suppose that
is a monoid and the element x in A has an inverse.
Then this inverse is unique.
Proof: ??

36. Properties of Groups Theorem (The cancellation laws): Let be
a group then
(i)
(ii)
Proof: (i) Suppose that
then
by axiom 3
a has an identity
and we have that
(ii) is proved similarly
Theorem (The division laws): Let be
a group then
(i)
(ii)
Proof ??

37. Theorem (double inverse) :If x is an element of
the group then
Proof:
Theorem (reversal rule)
If x and y are elements of the group then
Proof ??

38. For an arbitrary element of a groupwe
can define functions
and
such that
Theorem:
and
are permutations of G
Proof: Consider
[prove 1-1] suppose for x,y in G
[Prove onto] For any y in G
Corollary: In every row or column of the
multiplication table of G each element of G appears
exactly once.

39. Subgroups is a subgroup of the group if and is also a group Examples:
Test for a subgroup
Let H be a subset of G. Then
is a subgroup of
iff the following conditions all hold:
(1)
(2) H is closed under multiplication
(3)
For every group ,
and
are
subgroups
is called the trivial subgroup of
a proper subgroup of
is a subgroup
different from G
A non-trivial proper subgroup is a subgroup
equal neither to or to

40. Cosets Consider a set A with a subset H. Let .
Then the left coset of H with respect to a is
the set of elements:
This is denoted by
Similarly the right coset of H with respect to a is
and is denoted by
Example: Let A be the set of rotations
and
{0º,120º,240º}
. Let
then
which is the right coset with respect to

41. Normal Subgroups Let be a subgroup of . Then
is a normal subgroup if, for any
, the left
coset
is equal to the right coset
is a normal subgroup where
e.g.
Theorem: In an Abelian group, every subgroup
is a normal subgroup

42. Coset cardinality
Theorem: For any H subset of G and any a in G |a•H|=|H|
Proof:
By definition of Coset |a•H|≤|H|
Now suppose |a•H|<|H| then there must exist b and c distinct elements of H such that a•b=a•c.
But by the cancellation law this implies that b=c which is a contradiction.
Hence |a•H|=|H|

43. Coset partitioning
Theorem: Let a,b∈G and let H be a subgroup of G then either: a•H=b•H or: a•H∩ b•H=∅
Proof:
Suppose a•H∩ b•H≠∅ then there exist s and t in H such that a•s=b•t.
In this case a= b•t•s-1 and for an arbitrary x in H a•x= b•t•s-1•x
Now by the inverse axiom and closure, t•s-1•x∈H and hence b•t•s-1•x∈b•H, therefore a•x∈b•H so that a•H⊆b•H
Similarly we can show that b•H⊆a•H
Hence if the two cosets are not disjoint then b•H=a•H

44. LeGrange’s theorem
Theorem: Let H be a subgroup of finite group G, then the cardinality of H evenly divides the cardinality of G (i.e |H| | |G|)
Proof
Let |G|. Now for each element ai of G we can generate a coset ai•H.
Now notice that ai∈ai•H because since H is a subgroup, e∈H and ai•e= ai
Suppose there are m distinct cosets of H then picking one representative ai from each this means that: G= a1•H∪ a2•H ∪ a3•H … ∪ am•H

45. LeGrange’s theorem
Now by the previous theorem it follows that since these m cosets are distinct then they must be disjoint.
Hence,
|G|=|a1•H|+ |a2•H| + |a3•H| … + |am•H|
Also by the cardinality theorem for cosets they all have the same cardinality, namely |H|. Hence, |G|=m.|H| as required

46. Order of an element
Let i be the smallest integer such that ai=e where a is an element of group G and e is the identity element.
If i exists we call it the order of a.
Otherwise we say that a has infinite order.

47. Subgroup generated by an element
Theorem: For any element a of G with finite order the set: H={aj: for some integer j} is a subgroup of G.
Notice: if i is the order of element a then
ai=e
ai+1=e•a=a1
ai+2= a •a =a2
ai+n=an

48. Example Let σ=(1 2 3 4), a permutation of 4 elements
Then {σ, σ2, σ3, σ4} is a subgroup of the group of permutations of {1,2,3,4}
The order of σ is 4
[Work it out!]

49. Order of elements in finite groups
If the group G is finite then all elements of G have finite order:
For any a∈G, since G is finite there must exist i<j such that ai=aj
a•ai-1=a•aj-1 cancellation law implies ai-1=aj-1
Repeated application of the cancellation law gives a=aj-i+1
a•e=a•aj-i implies e=aj-i

50. Corollary of LeGrange
Theorem: The order of every element of a finite group G, divides the order of G
Proof...
Every element of G has finite order n and hence generates a subgroup of order n.
Hence by LeGrange’s theorem n divides |G|

51. Isomorphism
Two groups are isomorphic if there is a bijection of one onto the other which preserves the group operations i.e. if
and
are groups then a bijection
is an isomorphism provided
Example: Consider the group of matrices
of the form where under matrix
multiplication. This is isomorphic to the group
The mapping is
An isomorphism from a group onto itself is
called an automorphism.

52. Homomorphisms
The idea of isomorphic algebraic structures can be readily generalised by dropping the requirement that the functional mapping be a bijection.
Let
and
be two algebraic systems
then a homomorphism from to
is a functional mapping
such that
Example: consider the two structures
then f such that
is a homomorphism between
and

53. Algebraic Structures with two Operations
So far we have studied algebraic systems with one binary operation. We now consider systems with two binary operations.
In such a system a natural way in which two operations can be related is through the property of distributivity;
Let
be an algebraic system with two
binary operations
and
. Then the operation
is said to distribute over the operation if
and
Example:
distributes over +
distributes over
distributes over

54. Ring An algebraic system is called a ring if
the following conditions are satisfied:
(1)
is an Abelian group
(2)
is a semigroup
(3) The operation
is distributive over the
operation
Example:
is a ring since
is an Abelian group
is a semigroup
distributes over +

55. Examples of rings <Z, +, ×> is a ring because:
<Z, +> is an Abelian group.
<Z, ×> is a semigroup.
× distributes over +
The set {[ ],a,b є R} is a ring under matrix addition and multiplication
{0,1,…,n-1} is a ring under addition and multiplication mod n
0 a 0 b

56. Rings of polynomials
Let the set R[x] be the set of all polynomial of the form: anxn+…+ a2x2+ a1x1+a0
for some n, where an,…,a0 єR
Then R[x] is a ring under addition and multiplication of polynomials
In fact for any ring R you can construct a ring of polynomials R[x] over R

57. Special types of ring A commutative ring is a ring in which is
A ring with unity contains an element 1 such
that
(0 is the identity of )
Example: the ring of 2x2 matrices under matrix
addition and multiplication is a ring with unity.
The element 1=I=

58. Division rings
A division ring is a (not necessarily commutative) ring with unity, in which every element a not equal to 0 has an inverse a-1 such that a•a-1= a-1•a=1
The ring of complex matrices of the form: [ ]
a b -b a

59. Integral Domains and Fields
is an integral domain if it is a commutative
ring with unity that also satisfies the following
property;
is also an integral domain
is a field if:
(1)
is an Abelian group
(2)
is an Abelian group
(3) The operation
is distributive over the
operation
Example:The set of real numbers with respect to
+ and is a field.
is not a field. Why?

60. Galois fields
For a prime number p the set {0,1,…,p-1} is a field under modular addition and multiplication mod p
A field (like this one) with finite number of elements is called a Galois field.

61. A Field is an Integral Domain
Let
be a field then certainly
is a commutative ring with unity. Hence, it only
remains to prove that
Now suppose
then if x=0 the above
holds. Consider the case then where
Since
is an Abelian group then it
must contain an inverse to x,
, for which the
following holds
Now
Therefore y=0 as required

62. Properties of a ring Theorem: if is a ring. Then
Proof: as for previous argument
Let -x denote the inverse of x under
Theorem: if
is a ring then the following
hold
(i)
(ii)
Proof: (i)

63. (ii)
for both (i) and (ii) the symmetric cases are
proved similarly

64. Property of an integral domain
Theorem: suppose that elements a,b and c of
an integral domain satisfy and
then b=c.
Proof:

65. Subrings and subfield Subring
If (A,⊕,•) is a ring then (H,⊕,•) is a subring if H⊆A and
(H,⊕,•) is a ring
Subfield
If (A,⊕,•) is a field then (H,⊕,•) is a subfield if H⊆A and
(H,⊕,•) is a field
Examples: Z is a subring of R, R is a subfield of C

66. Ring morphisms
A morphism between rings (A,⊕,•) and (B,*,⊗) is a function f:A→B such that: ∀x,y∈A
f(x⊕y)=f(x)*f(y) and
f (x•y)=f(x)•f(y)
From these we have that
f(0)=0′ where 0′ is the zero of (B,*,⊗)
Also f(-x)=-f(x)

67. Special morphisms An injective ring morphism is called a monomorphism
2. A surjective ring morphism is called an epimorphism
3. A bijective ring morphism is called a isomorphism

68. Examples of morphisms
f(a) = a mod n, is an epimorphism (surjective ring morphism) between Z and {0,1,…,n-1}
For the ring of polynomials R(x), f(p)=p(j) is an epimorphism into C, where p(j) is obtained by substituting j for x in the polynomial p

69. Galois theorem
For every prime power pk (k=1,2,…) there is a unique (upto isomorphism) finite field containing pk elements denoted by GF(pk)
All finite fields have cardinality pk

70. Galois theorem: examples
GF(2)
+ | · | | | | | 0 1
GF(3)
+ | · | | | | | | | 0 2 1

71. Partial Orderings We have introduced formal structure governing
the properties of various sets of elements under
one or two binary operations.
These elements can also be ordered and restricted
by binary relations.
In this section we revise our understanding of
binary relations in a set and also introduce a
graphical notation for binary relations.
A relation R on a set A is a partial order if it
satisfies;
(1) R is reflexive
(2) R is antisymmetric
(3) R is transitive
The pair (A,R) is called a partially ordered set
or poset
Example: Set of reals R with the relation

72. Example: The relation
can be defined on a
Boolean algebra by;
(1) Thus from the idempotent law
we find that
and hence the relation is
reflexive.
(2) If
From the commutative law
and hence the relation is antisymmetric
(3) If
then

73. We can think of a relation as being represented by
the set of pairs of elements which satisfy the
relation.
In this case a partial ordering on A corresponds
to a subset B of AxA satisfying
Other examples of partial orderings:
Divisibility on N: We say that a divides b iff there
is some x in Z such that ax=b. If this divisibility
exists we write a|b. Divisibility is a partial order
on N.
Inclusion on a set of sets X

74. Graphical Representations
We can represent partial orderings graphically
by means of a directed graph where the nodes
are elements of A and the directed edges give
the partial order relations.
e.g. the graph a b c d
Denotes the partial ordering on
{a,b,c,d}
where

75. Graphical Representations of the Axioms
Reflexive: a
Antisymmetric: the following does not occur a b
Transitive: a b c

76. Example: Divisibility relation on{2, 3, 4, 6, 8, 9, 18}
2|4 4|8 2|8 2|6 3|6 3|9 9|18 6|18 3|18 2|18 2 4 8 6 3 9 18

77. Example: The collection of all subsets of {a,b,c}
{a,c}
{b,c}
{a}
{b}
{c}

78. Hasse Diagrams Notice that some of the diagrams in the previous
examples were messy and difficult to read having
many links.
We can simplify these diagrams by introducing
certain conventions.
The Hasse diagram of a partially order set is a
drawing of the points in the set (as nodes) and
some of the links of the graph of the order relation.
The rules for drawing the Hasse diagram of a partial
order are:
(1) Omit all links that can be inferred from
transitivity.
(2) Omit all loops
(3) Draw links without arrow heads
(4) Understand that all arrows would
point upwards

79. Here are Hasse diagrams for the two examples
given previously:
Divisibility: 18 8 9 4 6 3 2

80. Example: subsets
{a,b,c}
{a,b}
{a,c}
{b,c}
{a}
{b}
{c}

81. Incomparable Elements
Consider the Hasse diagram for divisibility on
{2,3,....,10} 10 8 6 9 4 5 3 7 2
Notice that 5 and 6 are not related in either direction
Similarly for 2 and 3
If neither R(a,b) or R(b,a) then a and b are
incomparable or not comparable

82. Linear or Total Order
A linear or total order on a set A is a partial order
on A in which every two elements are comparable 5 4 3 2

83. Maximal and Minimal Elements
A maximal element of A is any element t of
A such that
A minimal element of A is any element
b of A such that
Example:For the subset ordering {a,b,c} is the
maximal element and is the minimal element
For divisibility on {2,.....,10} the maximal
elements are 6, 7, 8, 9 and 10
and the minimal elements
are 2, 3, 5 and 7
The element 4 is neither maximal nor minimal

84. Upper Bounds and Lower Bounds
Let S be a subset of A then x in A is an upper
bound of S if
Similarly z in A is a lower bound of S if
An element u is the least upper bound of S if
u is an upper bound of S and for every x an
upper bound of S R(u,x)
An element l is the greatest lower bound of S if
l is an upper bound of S and for every z a
lower bound of S R(z,l)
The least upper bound (lub) of S is sometimes
referred to as the supremum of S (sup S)
The greatest lower bound (glb) of F is sometimes
referred to as the infimum of S (inf S)

85. Lattices A partially ordered set in which every pair of
elements has a least upper bound and a greatest
lower bound is called a lattice. a b c d f e
This is not a lattice since {c,d} has no lub or
glb.
A lattice in which every subset has a lub and glb
is called complete.
Every finite lattice is complete.
For a complete lattice the lub of the whole lattice
is call top and the greatest lower bound bottom

86. Example: Consider elements of the form (a,b,c)
where a,b and c can take the values 0 or 1. For
two such elements f and g we say that
if each coefficient of f is less than or equal to
the corresponding coefficient of g
e.g.
but not
(111)
(101)
(011)
(110)
(001)
(100)
(010)
(000)

87. Meet and Join In a lattice the following equations
define binary operations on A
is called the meet operation and is
called the join operation.
They have the following properties
Commutativity:
Associativity:
Since
is an upper bound of a and b
Similarly for the meet

88. Theorem
and If
then
Proof
Let 1 denote the lub of the whole lattice and 0
denote the glb of the whole lattice. Then

89. Example Let us order the following set of numbers with the
operation “is a factor of”. A={3,9,12,15,36,45} 45 36 9 15 12 3
The join operation
is the least common multiple
The meet operation
is the greatest common divisor

90. Complemented Lattice For a complemented lattice we have that for
there exists
such that:
e.g. 1 a b c

91. Distributive Lattice A lattice is distributive if:
e.g. the following lattice is not distributive e d b c a
Since

92. Boolean Algebra A Boolean Algebra consists of two binary
operations and and the unary operation
on a set B with distinct elements 0 and 1 such
that the following hold.
(1) The commutative laws:
(2) The associative laws:
(3) The Distributive laws:
(4) The Identity Laws:

93. (5) The Complementation Laws:
Theorem If
is a complemented distributive lattice then
is a Boolean algebra where
correspond to the meet, join and complement
operations on L respectively
Proof ?

94. (6) The following Idempotent Laws can be derived:
Proof
(7) The following Identity Laws can also be derived
Proof