Relations and Functions-, Types of Functions

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• A pair of objects, written in a specified order is called an ordered pair.

• A pair of objects, written in a specified order is called an ordered pair.

• For the ordered pair (a,b), a is called the first element and b, the second element.

• The ordered pairs (a,b) and (c,d) are equal if their corresponding elements are equal.

Symbolically, (a,b) = (c,d) if a=c, b=d.

•  The cartesian product of two non-empty sets A and B is defined as the set of all ordered pairs (a, b), where a Î A b Î B and is denoted by A x B.

Symbolically,

• Relation from a set A to a set B is a subset of A x B.

- A relation R on a set A is reflexive if a R a, " a Î A- A relation R on a set A is symmetric if a R b Þ b R a.

- A relation R on a set A transitive if a R b and b R c Þ a R c.

• A relation R is said to be an equivalence relation on A if it is reflexive, symmetric and transitive.

• A relation f from A to B is said to be a function if every element of A occurs as the first co-ordinate of at least one ordered-pair in f and f does not contain two ordered pairs having the same first co- ordinate.

• Let A and B be two non-empty sets. A correspondence between the elements of A and B is called a function from A to B if to each element of A, there corresponds exactly one element of B.

• Let f : A → B be a function, then:

(a) A is called the domain of f

(b) B is called the co-domain of f

(c) The set {f(x) : x Î A} is called the range of f. Range (f) is a subset of B

• A function f : A→ B is said to be one-one or injective if f(x₁) = f(x₂) Þ x₁ = x₂.

• A function f : A→ B is said to be onto or surjective if every bÎB is the image if some a Î A.

• A function f : A→ B is bijective if f is one-one and onto.

• A function f : X→ Y is called

Here p(x) and q(x) are real polynomials.

• (i) Every function from A to B is a relation from A to B

(ii) A relation from A to B is a function from A to B only when

(a) each element of A is the first element of some element in R and

(b) no element of A is the first element of any two distinct elements in R.

• A function f : A→B is called a one-one function if the images of distinct elements of A are also distinct elements of B

         Symbolically

         

A one-one function is also called an injective function.

• f : A→B is not 1 - 1 if there exists two distinct elements in A, whose images are same. If f : A→B is not 1 - 1 then it is said to be a many - one function.

• A function f : A→B is called an onto function if every element of B is the image of at least one element of A

An onto function is also called a subjective function.

• If f : A→B is a onto function, then range (f) = B.

          If A→B is not onto, then there exists at least one element in B which is not the image of any element in A.

          If f : A→B is not onto, then it said to be an into function.

• A function f : A→ B is called a one-one, onto function if it is both one-one and onto.

• (i) Let f : A→B and g : B→C be two functions. The function gof : A→C defined by (gof)(a) = g(f(a)), a ÎA is called the composite function of f and g.

          (ii) If f and g be functions from A to A then fog and gof are both defined. In general fog  ¹ gof.

• A function from A x A into A is called a binary operation on A.