Page 11 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Obviously, f (x) B. Notes
If A and B are any two non-empty sets, then a function f from A to B is a subset f of A × B satisfying
the following condition:

(i) a A, (a, b) f for some b B;
(ii) (a, b) f and (a, b ) f b = b
The first condition ensures that each element in A will have image. The second condition
guarantees that the image is unique.
If the domain and co-domain of a function f are both the same set say f : A A, then f is often
called an Operator or Transformation of A.

Two functions f and g of A B are said to be equal iff f (x) = g (x) x A and we write f = g. For
two unequal mappings from A to B, there must exist at least one element x A such that
f (x) g (x).

Types of Functions

If the function f : A B is such that there is at least one element in B which is not the f-image of
any element in A, then we say that f is a function of A ‘into’ B. In this case the range of f is a proper
subset of the co-domain of f.
If the function f : A B is such that each element in B is the f-image of at least one element in A,
then we say that f is a function of A ‘onto’ B. In this case the range of f is equal to the co-domain of
f, i.e., f (A) = B. Onto mapping is also sometimes known as surjection.
A function f : A B is said to be one-one or one-to-one if different elements in A have different
f-images in B, i.e., if
f (x) = f (x ) x = x (x and x A).
One-to-one mapping is also sometimes known as injection.
A mapping f : A B is said to be many-one if two (or more than two) distinct elements in A have
the same f-image in B.
If f : A B is one-one and onto B, then f is called a one-to-one correspondence between A and B.
One-one onto mapping is also sometimes known as bijection.
Two sets A and B are said to be have the same number of elements iff a one-to-one correspondence
of A onto B exists. Such sets are said to be cardinally equivalent and we write A ~ B.

Let A be any set. Let the mapping f : A A be defined by the formula f (x) = x, x A, i.e. each
element of A be mapped on itself. Then f is called the identity mapping on A. We shall denote
this function by I .
A
Inverse Mapping

Let f be a function from A to B and let b B. Then the inverse image of the element b under f
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denoted by f (b) consists of those elements in A which have b as their f-image.
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Let f : A B be a one-one onto mapping. Then the mapping f : B A, which associates to each
element b B, the element a A, such that f (a) = b is called the inverse mapping of the mapping
f : A B.
It must be noted that the inverse mapping of f : A B is defined only when f is one-one onto, and
it is easy to see that the inverse mapping f : B A is also one-one and onto.
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