Page 10 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Two results which interrelate union and intersection of sets with their complements are as
follows:
(i) the complement of the union is intersection of the complements, i.e.,

(A B) = A B , and
(ii) the complement of the intersection is the union of the complements, i.e.,
(A B) = A B .
Suppose A and B are two sets. Then the set (A – B) (B – A) is called the symmetric difference of
the set A and B and is denoted by A B.
Since (A – B) (B – A) = (B – A) (A – B)
A B = B A.

Product Set

Let A and B be two sets, a A and b B. Then (a, b) denotes what we may call an ordered pair. The
element a is called the first coordinate of the ordered pair (a, b) and the element b is called its
second coordinate.

If (a, b) and (c, d) are two ordered pairs
then (a, b) = (c, d) iff a = c and b = d.
If A and B are two sets, the set of all distinct ordered pairs whose first coordinate is an element
of A and whose second coordinate is an element of B is called the Cartesian product of A and B (in
that order) and is denoted by A × B. Symbolically,
A × B = {(a, b) : a A and b B}.

In general A × B B × A. If A has n elements and B has m elements, then the product set A × B has
nm elements. If either A or B is a null set, the A × B = . If either A or B is infinite and the other is
not empty, the A × B is infinite.
We may generalise the definition of the product sets. Let A , A , …, A be n given sets. The set of
1 2 n
ordered n-tuples (a , a , …, a ) where a A , a A , …, a A is called the Cartesian product of
1 2 n 1 1 2 2 n n
A , A , …, A and is denoted by A × A × A × … × A .
1 2 n 1 2 3 n
Functions or Mappings

Let A and B be two given sets. Suppose there is a correspondence, denoted by f, which associates
to each members of A, a unique member of B. Then f is called a function or a mapping from A to B.
The mapping f from A to B is denoted by

f
f : A B or by A B.
Suppose f is a function from A to B. The set A is called the domain of the function f and B is called
the co-domain of f. The element y B which the mapping f associates to an element x A is
denoted by f (x) and is called the f-image of x or the value of the function f for x. The element x
may be referred to as a pre-image of f (x). Each element of A has a unique image and each
element of B need not appear as the image of an element in A. There can be more than one
element of A which have the same image in B. We define the range of f to consist of those
elements of B which appear as the image of at least one element in A. We denote the range of
f : A B by f (A). Thus
f (A) = {f (x) : x A}.

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