Page 8 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes One should note carefully the difference between a collection and a set. Every collection is not a
set. For a collection to be a set, it must be well defined. For example the collection of “any four
natural numbers” is not a set. The members of this collection are not well defined. The natural
number 5 may belong or may not belong to this collection. But the collection of “the first four
natural numbers” is a set. Obviously, the members of the collection are well-defined. They are
1, 2, 3 and 4.
A set is usually denoted by a capital letter, such as A, B, C, X, Y, Z etc. and an element of a set by
the small letter such as a, b, c, x, y, z etc.
A set may be described by actually listing the objects belonging to it. For example, the set A of
single digit positive integers is written as
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Here the elements are separated by commas and are enclosed in brackets { }. This is called the
tabular form of the set.
A set may also be specified by stating properties which its elements must satisfy. The set is then
described as follows:

A = {x : P(x)} and we say that A is the set consisting of the elements x such that x satisfies the
property P(x). The symbol “.” is read “such that”. Thus the set X of all real numbers is simply
written as
X = {x : x is real} = {x | x is real}.
This way of describing a set is called the set builder form of a set.
When a is an element of the set A, we write a A. If a is not an element of A, we write a A.

When three elements, a, b and c, belong to the set A, we usually write a, b, c A, instead of
writing a A, b A and c A.
Two sets A and B are said to be equal iff every element of A is an element of b and also every
element of B is an element of A, i.e. when both the sets consist of identical elements. We write
“A = B” if the sets A and B are equal and “A B” if the sets A and B are not equal.

If two sets A and B are such that every element of A is also an element of B, then A is said to be
a subset of B. We write this relationship by writing A B.
If A B, then B is called a superset of A and we write B A, which is read as ‘B’ is a super-set of
A’ or ‘B contains A’.
If A is not a subset of B, we write A B, which is read as ‘A is not a subset of B’. Similarly B A
is read as ‘B is not a superset of A’.
From the definition of subset, it is obvious that every set is a subset of itself, i.e., A A. We call
B a proper subset of A if, first, B is a subset of A and secondly, if B is not equal to A. More briefly,
B is a proper subset of A, if
B A and B A.
Another improper subset of A is the set with no element in it. Such a set is called the null set or
the empty set, and is denoted by the symbol . The null set is a subset of every set, i.e., A.
If A is any set, then the family of all the subsets of A is called the power set of A. The power set of
A is denoted by P(A). Obviously and A are both elements of P(A). If a finite set A has n
n
elements, then the power set of A has 2 elements.

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