Page 12 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Product or Composite of Mappings

Let f : X Y and g : Y Z. Then the composite of the mappings f and g denoted by (g o f), is a
mapping from X to Z given by (g o f) : X Z such that (g o f) (x) = g [f (x)], x X.
If f : X X and g : X X then we can find both the composite mappings g o f and f o g, but in
general f o g g o f.
The composite mapping possesses the following properties:
(i) The composite mapping g o f is one-one onto if the mappings f and g are such.

(ii) If f : X Y is a one-one onto mapping,
then f o f = I and f o f = I .
–1
–1
y x
(iii) If f : X Y and g : Y Z are two one-one onto mappings, and f : Y X and g : Z Y
–1
–1
are their inverses, then the inverse of the mapping g o f : X Z is the mapping f o g : Z
–1
–1
X.
(iv) If f : X Y, g : Y Z, h : Z U be any mappings, then h o (g o f) and (h o g) o f are equal
mappings of X into U, i.e. the composite mapping is associative.
Relation
If a and b be two elements of a set A, a relation R between them, is symbolically written as aRb,
which means a in R—related to b.

For example, if R is the relation >, the statement a R b means a is greater than b.
A relation R is said to be well defined on the set A if for each ordered pair (a, b), where a, b A,
the statement a R b is either true or false. A relation in a set A is a subset of the product set A × A.

Inverse Relation

Let R be a relation from A to B. The inverse relation of R denoted by R , is a relation from B to
–1
A defined by

x
y
R 1 {( , ): y B ,x A ,( , ) A B }
x
y
Clearly, if R is a relation from A to B, then the domain of R is identical with the range of R and
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the range of R is identical with the domain of R .
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Difference between Relations and Functions
Suppose A and B are two sets. Let f be a function from A to B. Then by the definition of function
f is a subset of A × B in which each a A appears in one and only one ordered pair belonging to
f. In other words f is a subset of A × B satisfying the following two conditions:
(i) for each a A, (a, b) f for some b B,
(ii) if (a, b) f and (a, b ) f, then b = b .
On the other hand every subset of A × B is a relation from A to B. Thus every function is a relation
but every relation is not a function. If R is a relation from A to B, then domain of R may be a subset
of A. But if f is a function from A to B, then domain of f is equal to A. In a relation from A to B an
element of A may be related to more than one element in B. Also there may be some elements
of A which may not be related to any element in B. But in a function from A to B each element of
A must be associated to one and only one element of B.

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