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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:
Types of 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.
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.
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