01) The number of elements in a set:

The Number of distinct elements in a finite set A is denoted by n (A).

02) Some Important Results:

Cartesian Product of two sets:

Relation:

If A and B are any two non-empty sets, then any subset of A x B is called relation from A to B. If R is the relation from A to B and (x, y) belongs to R, then we say that xRy. Here Set A is the Domain, set B is the Co-Domain. Y is called the image of x under relation R. Similarly, x is called the pre-image of y under relation R.

Range:

Set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Relation R.

Function:

A and B are two non-empty sets. The Function f from A to B is said to be a function if every element of set A is associated with the unique element of set B and is denoted as f: A -> B. and we say that y = f(x). Here y is the image of x under function f.

Here we say that A is the Domain and B is Co-Domain.

Range:

Set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Function f.

Onto Function:

f: A-> B is Onto function if every element of Co-Domain B is associated with some

elements of A under Function f. (Note: Range of function "f" is the same as Co-Domain).

Into Function:

f: A ->; B is Into function if there exists at least one element extra in Co-Domain B which is not associated with any element of Domain A under Function f. (Note: Range of function "f" is not same as Co-Domain).

Even Function:

f : A -> B is said to be Even Function if f (-x) = f (x).

Example: f (x) = cos x

Example: f (x) = x

Odd Function:

f : A -> B is said to be Even Function if f (-x) = - f (x).

Example: f (x) = sin x

Example: f (x) = x

If f: A ->; B, g : B-> C, then gof : A -> C where gof (x) = g[f(x)]

Inverse Function:

If f : A -> B is one-one onto Function, then g : B ->A is known as an inverse function of f and is denoted as f

Anil Satpute

The Number of distinct elements in a finite set A is denoted by n (A).

02) Some Important Results:

Cartesian Product of two sets:

Relation:

If A and B are any two non-empty sets, then any subset of A x B is called relation from A to B. If R is the relation from A to B and (x, y) belongs to R, then we say that xRy. Here Set A is the Domain, set B is the Co-Domain. Y is called the image of x under relation R. Similarly, x is called the pre-image of y under relation R.

Range:

Set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Relation R.

Function:

A and B are two non-empty sets. The Function f from A to B is said to be a function if every element of set A is associated with the unique element of set B and is denoted as f: A -> B. and we say that y = f(x). Here y is the image of x under function f.

Here we say that A is the Domain and B is Co-Domain.

Range:

Set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Function f.

Onto Function:

f: A-> B is Onto function if every element of Co-Domain B is associated with some

elements of A under Function f. (Note: Range of function "f" is the same as Co-Domain).

Into Function:

f: A ->; B is Into function if there exists at least one element extra in Co-Domain B which is not associated with any element of Domain A under Function f. (Note: Range of function "f" is not same as Co-Domain).

Even Function:

f : A -> B is said to be Even Function if f (-x) = f (x).

Example: f (x) = cos x

Example: f (x) = x

^{2}Odd Function:

f : A -> B is said to be Even Function if f (-x) = - f (x).

Example: f (x) = sin x

Example: f (x) = x

^{3}
Composite Function:

If f: A ->; B, g : B-> C, then gof : A -> C where gof (x) = g[f(x)]

Inverse Function:

If f : A -> B is one-one onto Function, then g : B ->A is known as an inverse function of f and is denoted as f

^{-1 }: B -> A.Anil Satpute