B tech. Discrete mathematics (I. T & Comp. Science Engg.) Syllabus

EXAMPLE:

Determine whether (P(S ),  ) is a lattice where S is a set.

Solution
Let A and B be two subsets of S . The least upper bound and the greatest lower bound of A and B are A U B and A ∩B, respectively.

Hence (P(S ),  ) is a lattice.

CODES AND GROUP CODES

INTRODUCTION :

In today’s modern world of communication, data items are constantly being transmitted from point to point.

Different devices are used for communication. The basic unit of information is message. Messages can be represented by sequence of dots and das

Let

B = {0,1}be the set of bits. Every character or symbol can be

represented by sequence of elements of B. Message are coded in O’s and 1’s and then they are transmitted. These techniques make use of group theory. We will see a brief introduction of group code in this chapter. Also we will see the detection of error in transmitted message.

^{B } ^0,1 ^{is a group under the binary operation  whose}
The set table is as follows :

	0	1
0	0	1
1	1	0

We have seen that B is a group.

It follows from theorem - “If G₁ and G₂ are groups then

G  G₁  G₂ is a group with binary operation defined by

_a₁, b_{1 }a₂, b_{2 }  _a₁, a₂, b₁, b_{2 }. So B^m  B B   B (m factors) is

a group under the operation  defined by

_x₁, x_{2 ....}x_{m }  _y₁, y₂  y_{m }  _x₁  y₁, x₂  y₂, x_m  y_{m }

observe that B^m has 2^m elements. i.e. order of group B^m is 2^m.