Relations and Functions-II, Equivalence Relation, Classification of Function

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Equivalence Relation

A relation on a set is said to be an equivalence relation on iff

(i) It is reflexive i.e. for all

(ii) It is symmetric i.e. for all

(iii) It is transitive i.e. and for all

For example the relation ‘is congruent to’ is an equivalence relation because

(i) it is reflexive as for all where is a set of triangles.

(ii) it is symmetric as

(iii) it is transitive as and

it means and

Example:

Show that the relation defined on the set of all triangles in a plane as

is similar to is an equivalence relation.

Solution:

We observe the following properties of relation ;

Reflexivity:

We know that every triangle is similar to itself. Therefore, for all is reflexive.

Symmetricity:

Let , then

is similar to

is similar to

, So, is symmetric.

Transitivity:

Let such that and .

Then and

is similar to and is similar to

is similar to

Hence, is an equivalence relation.

Classification of Function:

Let f be a function from to . If every element of the set is the image of at least one element of the set i.e. if there is no unpaired element in the set then we say that the function of maps the set onto the set . Otherwise we say that the function maps the set into the set .

Functions for which each element of the set is mapped to a different element of the set are said to be one-to-one.

One-to-one function

One-to-one function

One-to-One Function

One-to-one function

The domain is

The co-domain is

The range is

A function can map more than one element of the set to the same element of the set . Such a type of function is said to be many-to-one.

Many-to-one function

Many-to-one function

Many-to-One Function

Many-to-one function

The domain is

The co-domain is

The range is

A function which is both one-to-one and onto is said to be a bijective function.

One-to-one and onto Function 1

One-to-One and Onto Function 1

One-to-one and onto Function 1

  1. (b)

One-to-one and onto Function 2

One-to-One and Onto Function 2

One-to-one and onto Function 2

(c) (d)

Fig. (a) shows a one-to-one function mapping into .

Fig. (b) shows a one-to-one function mapping onto .

Fig. (c) shows a many-to-one function mapping into .

Fig. (d) shows a many-to-one function mapping onto .

Function shown in Fig. (b) is also a bijective Function.

Example:

Prove that defined by is neither one-one nor onto function.

Solution:

We have domain giving

or or

or is not one-one function.

Again let where codomain

domain.

no real value of x in the domain.

is not an onto function.

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