Sets, Relations And Functions

Introduction to functions:

Function is defined as a rule or a manner or a mapping or a correspondence f which maps each & every element of set A with a unique element of set B. It is denoted by:

F: A → B or   we read it as “ f is a function from A to B”

Figure -2 does not represent a function because conversion is allowed (figure -3) But diversion is not allowed.

DOMAIN, CO-DOMAIN & RANGE OF A FUNCTION

Let f: A → B, then the set A is known as the domain of f & the set B is known as co-domain of f. If a member 'a' of A is associated to the member 'b' of B, then'b' is called the f-image of 'a' and we write b = f(a). Further 'a' is called a pre-image of `b'. The set {f(a): " a ∊ A} is called the range of f and is denoted by f(A). Clearly f(A) Í B.

If only expression of f(x) is given (domain and codomain are not mentioned), then domain is set of those values of 'x' for which f (x) is real, while codomain is considered to be  (except in ITFs)

 A function whose domain and range are both subsets of real numbers is called a real function.

Algebraic Operations on Functions: If f & g are real valued functions of x with domains A and B respectively, then both f & g are defined in A Ç B. Now we define f+g, f-g, (f. g) & (f/g) as follows:

1.  (f±g)(x) = f(x) ± g(x)
2. (f.g) (x) = f(x).g(x)] domain in each case is A Ç B
3.   Domain is {x|x ∊ A ÇB and g (x) ¹ 0}.

Types of function:

• A real function has the set of real numbers or one of its subsets both as its domain and as its range.
•  Identity function: f: X ® X is an identity function if f(x) = x for each x Î A
• Constant function: A constant function is one which maps each element of the domain to a constant. Domain of this function is R and range is the singleton set {c}, where c is a constant.
• Polynomial function: f: R ® R defined as y = f(x) = a0 + a1x + a2x 2 + …. + an x n , where n is a nonnegative integer and a0, a1, a2, …an Î R.
• Modulus function: f: R ® R denoted by f(x) = |x| for each x Î R. The modulus function is defined as f(x) = x if x ³ 0 and f(x) = −x if x < 0
• Step or greatest integer function: A function f: R ® R denoted by f(x) = [x], x Î R, where [x] is the value of greatest integer, less than or equal to ‘x’ is called a step or greatest integer function. It is also called a floor function.
• Smallest integer function: A function f: R ® R denoted by f x x () = éù êú , x Î R where éù x êú is the value of the smallest integer, greater than or equal to ‘x’ is called a smallest integer function. It is also known as the ceiling function.
• Signum function: f(x)= |x| x , x ¹ 0 and 0 for x = 0. The domain of a signum function is R and its range is {−1, 0, 1}.  ,
• If ‘a’ is a positive real number other than unity, then a function which relates each x R Î to x a is called the exponential function.
• If a > 0 and a ¹ 1, then the function defined by f(x) = log_a x, x > 0 is called the logarithm function.
• The function defined by f: R {0} → R such that, f(x) =  is called the reciprocal  function.
• The function defined by f:R⁺ → R such that, f(x) =  is called the square root function.
• The function defined by f: R → R such that, f(x) = x²  is called the square function.
• The function defined by f: R → R such that, f(x) = x^1/3 is called the cube root function.     

Introduction to Sets, Relations: 

To understand relations basic knowledge of sets is required. A set is a well-defined collection of objects. Similar to Sets Relations can be represented in two ways- Roster or Tabular form and Set Builder form.

• Roster form: All the elements of a set are listed and separated by commas and are enclosed within braces { }. Elements are not repeated generally.
• Set Builder form: In set builder form, a set is denoted by stating the properties which its members satisfy. A set does not change if one or more elements of the set are repeated.

An empty set is the set having no elements in it. It is denoted by f or { }. A set having a single element is called a singleton set. On the basis of the number of elements, sets are of two types—finite and infinite sets. A finite set is a set in which there are a definite number of elements. Now, f or { } or null set is a finite set as it has 0 number of elements, which is a definite number. A set which is not finite is called an infinite set. All infinite sets cannot be described in the roster form. Two sets are equal if they have exactly the same elements. Two sets are said to be equivalent if they have the same number of elements.

If A Í B and B Í A, then A = B

Null set f is the subset of every set including the null set itself.