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VTU BE CSE 3rd sem notes

Discrete Mathematical Structures (DMS)

Chapter 1
Sets and Notation
1.1 Defining sets
Definition. A set is an unordered collection of distinct objects. The objects in a
set are called the elements, or members, of the set. A set is said to contain its elements.
A set can be defined by simply listing its members inside curly braces. For example,
the set {2, 4, 17, 23} is the same as the set {17, 4, 23, 2}. To denote membership we
use the 2 symbol, as in 4 2 {2, 4, 17, 23}. On the other hand, non-membership is
denoted as in 5 62 {2, 4, 17, 23}.
If we want to specify a long sequence that follows a pattern, we can use the ellipsis
notation, meaning “fill in, using the same pattern”. The ellipsis is often used after two
or more members of the sequence, and before the last one, as follows: {1, 2, . . . , n}.
The pattern denoted by the ellipsis should be apparent at first sight! For instance,
{1, . . . , n} is generally regarded as underspecified (that is, too ambiguous). Of course,
even {1, 2, . . . , n} is still ambiguous—did we mean all integers between 1 and n, all
powers of two up to n, or perhaps the set {1, 2, 25, n}?—but is generally sufficient,
unless you really do mean all powers of two up to n, in which case {20, 21, 22, . . . , 2k} for an appropriate k is a better choice. The ellipsis can also be used to define an
infinite set, as in the following.
Definition. The set of natural numbers or nonnegative integers, denoted by N, is
defined as {0, 1, 2, . . .}.
To avoid ambiguities it is often useful to use the set builder notation, which lists
on the right side of the colon the property that any set element, specified on the left
side of the colon, has to satisfy. Let’s define the positive integers using the set builder
notation

N+ = {x : x 2 N and x > 0}.
We can also write
N+ = {x 2 N : x > 0}.

This is a matter of taste. In general, use the form that will be easiest for the reader
of your work to understand. Often it is the least “cluttered” one.
Ok, now onto the integers:
Z = {x : x 2 N or −x 2 N}.
Hmm, perhaps in this case it is actually better to write
Z = {. . . ,−2,−1, 0, 1, 2, . . .}.
Remember, when you write mathematics, you should keep your readers’ perspective
in mind. For now, we—the staff of this course—are your readers. In the future it
might be your colleagues, supervisors, or the readers of your published work. In
addition to being reasonably formal and unambiguous, your mathematical writing
should be as clear and understandable to your intended readership as possible.
Here are the rational numbers:



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