**Ordered Pair**

An ordered pair consists of two objects or elements in a given fixed order.

**Equality of Two Ordered Pairs**

Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

**Cartesian Product of Two Sets**

For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the cartesian product of sets A and B and is denoted by A × B.

Thus, A × B = {(a, b) : a ∈ A and b ∈ B}

If A = Φ or B = Φ, then we define A × B = Φ

Note:

- A × B ≠ B × A
- If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
- If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.

**Relations**

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set 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 set of all first elements in a relation R is called the domain of the relation B, and the set of all second elements called images is called the range of R.

Note:

- A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram which is a visual representation of relation.
- If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2
^{mn}

**Inverse of Relation**

For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R, denoted by R^{-1} is a relation from B to A and it is defined by

R^{-1} ={(b, a) : (a, b) ∈ R}

Domain of R = Range of R^{-1} and

Range of R = Domain of R^{-1}.

**Functions**

A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set B.

In other words, a function f is a relation such that no two pairs in the relation have the first element.

**Real-Valued Function**

A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both are subsets of R, then f is called a real function.

**Some Specific Types of Functions**

Identity function: The function f : R → R defined by f(x) = x for each x ∈ R is called identity function.

Domain of f = R; Range of f = R

**Constant function:** The function f : R → R defined by f(x) = C, x ∈ R, where C is a constant ∈ R, is called a constant function.

Domain of f = R; Range of f = C

**Polynomial function:** A real valued function f : R → R defined by f(x) = a_{0} + a_{1}x + a_{2}x^{2}+…+ a_{n}x^{n}, where n ∈ N and a_{0}, a_{1}, a_{2},…….. a_{n} ∈ R for each x ∈ R, is called polynomial function.

**Rational function:** These are the real function of type , where f(x)and g(x)are polynomial functions of x defined in a domain, where g(x) ≠ 0.

**The modulus function:** The real function f : R → R defined by f(x) = |x|

or

for all values of x ∈ R is called the modulus function.

Domaim of f = R

Range of f = R^{+} U {0} i.e. [0, ∞)

**Signum function:** The real function f : R → R defined

by f(x) = , x ≠ 0 and 0, if x = 0

or

is called the signum function.

Domain of f = R; Range of f = {-1, 0, 1}

**Greatest integer function:** The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function.

Domain of f = R; Range of f = Integer

**Fractional part function:** The real function f : R → R defined by f(x) = {x}, x ∈ R is called the fractional part function.

f(x) = {x} = x – [x] for all x ∈R

Domain of f = R; Range of f = [0, 1)

**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.

**Subtraction of a real function from another:** 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.

**Multiplication by a scalar:** Let f : X → R be a real function and K be any scalar belonging to R. Then, the product of Kf is function from X to R defined by (Kf)(x) = Kf(x) for all x ∈ X.

**Multiplication of two real functions:** Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, product of these two functions i.e. f.g : X → R is defined by (fg) x = f(x) . g(x) ∀ x ∈ X.

**Quotient of two real functions:** Let f and g be two real functions defined from X → R. The quotient of f by g denoted by is a function defined from X → R as