Chap ter 0
Intro duc tion
0.1 BASICS
0.1.1 Sets
A “set” is a collection of objects. For example, the collection of four letters a,
b, c and d is a set, which is written as
L = { a, b, c, d }
The objects comprising a set are called its “elements” or “members”.
A set having only one element is called a “singleton”. A set with no
element at all is called the “empty set”, which is denoted by ∅.
It is essential to have a criterion for determining, for any given thing,
whether it is or is not a member of the given set. This criterion is called the
“Membership criterion” of the set.
There are two common ways to indicate the members of a set:
(i) List all the elements, e.g, {a, e, i, o, u}.
(ii) Provide some kind of an algorithm or a rule, such as a grammar.
Let us now take a look at the nota tion that is being used to denote sets.
(a) To indicate that x is a member of the set S, we write x ∈ S.
(b) If every element of set A is also an element of set B, we say that A
is a “subset” of B, and write A ⊆ B.
(c) If every element of set A is also an element of set B, but B also has
some elements not contained in A, we say that A is a “proper
subset” of B and write A ⊂ B.
(d) We denote the “empty set” as { } or ∅.
The set oper a tions are as described below.
(a) Union
The “union” of two sets is the set that has objects that are elements of at least
one of the two given sets, and possibly both.
Introduction