maths

RELATIONS & FUNCTIONS
CONCEPTS AND RESULTS
** Cartesian Products of Sets : Given two non-empty sets P and Q. The cartesian product P × Q is
the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p , q) : p  P, q  Q }
** Two ordered pairs are equal, if and only if the corresponding first elements, are equal and the
second
elements are also equal.
** If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e.
if n(A) = p and n(B) = q, then n(A × B) = pq.
** If A and B are non-empty sets and either A or B is an infinite set, then so is A × B.
** A × A × A = {(a, b, c) : a, b, c  A}. Here (a, b, c) is called an ordered triplet.
** Relation : A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian
product
A × B. The subset is derived by describing a relationship between the first element and the second
element of the ordered pairs in A × B. The second element is called the image of the first element.
** The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the
domain of the relation R.
** The set of all second elements in a relation R from a set A to a set B is called the range of the
relation
R. The whole set B is called the co-domain of the relation R. Range  co-domain.
** A relation may be represented algebraically either by the Roster method or by the Set-builder
method.
** An arrow diagram is a visual representation of a relation.
** The total number of relations that can be defined from a set A to a set B is the number of possible
subsets of A × B. If n(A ) = p and n(B) = q, then n (A × B) = pq and the total number of relations is
pq
2 .
** A relation R from A to A is also stated as a relation on A.

** Function: A relation f from a set A to a set B is said to be a function if every element of set A has
one and only one image in set B.
In other words, a function f is a relation from a non-empty set A to a non-empty set B such that
the domain of f is A and no two distinct ordered pairs in f have the same first element.
If f is a function from A to B and (a, b)  f, then f (a) = b, where b is called the image of a under
f and a is called the pre-image of b under f.
** A function which has either R or one of its subsets as its range is called a real valued function.
Further,
if its domain is also either R or a subset of R, it is called a real function.
Some functions and their graphs

** Identity function Let R be the set of real numbers. Define the
real valued function f : R →R by y = f(x) = x for each x  R.
Such a function is called the identity function. Here the domain
and range of f are R.




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, **Constant function : Define the function f: R → R by y = f (x) = c, x  R
where c is a constant and each x  R. Here domain of f is R and its range
is {c}.




**Polynomial function : A function f : R→R is said to be polynomial function if for each x in R,
y = f (x) = a0 + a1x + a2x2 + ...+ an xn, where n is a non-negative integer and a0, a1, a2,...,an  R.
f (x)
** Rational functions : are functions of the type , where f(x) and g(x) are polynomial functions
g(x )
of x defined in a domain, where g(x) ≠ 0.

** The Modulus function : The function f: R→R defined by
f(x) = |x| for each x  R is called modulus function. For each
non-negative value of x, f(x) is equal to x.
But for negative values of x, the value of f(x) is the negative of
 x, x  0
the value of x, i.e., f ( x )   .
 x, x  0




** Signum function : The function f : R→R defined by
 1, if x  0

f ( x )   0, if x  0
 1, if x  0

is called the signum function. The domain of the
signum function is R and the range is the set {–1, 0, 1}.




** Greatest integer function :
The function f : R → R defined by f(x) = [x], x  R assumes
the value of the greatest integer, less than or equal to x.
Such a function is called the greatest integer function.
[x] = –1 for –1 ≤ x < 0
[x] = 0 for 0 ≤ x < 1
[x] = 1 for 1 ≤ x < 2
[x] = 2 for 2 ≤ x < 3 and so on.


Algebra of real functions
** Addition of two real functions : Let f : X → R and g : X → R be any two real functions, where X
 R.
Then, we define (f + g): X → R by (f + g) (x) = f (x) + g (x), for all x  X.


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