Sets , Relations and Functions

Sets
A set is a collection of elements. There are various forms of set declaration e.g. roster method, set builder method or direct method.

Union

If A and B are two sets then the collection of all elements of A as well as B, is called as A Union B.

Intersection

If A and B are two Sets then the collection of elements common to both, is called as A intersection B.

Venn Diagram

Sets Formulae

De Morgan's Theorem

Relations
A relation is used to describe certain properties that connect two sets. Basically, a relation is linkage between elements of two sets , based on certain properties.

Properties
Reflexive

A->A

Transitive

If A->B, B->C then A->C

Symmetric

If A->B, then B->A

Equivalence Relation

A relation that is reflexive, symmetric and transitive.

Function
A function defines the behavior of y co-ordinate w.r.t. x co-ordinate. A function is represented by f(x). f(x) stands for the value of y co-ordinate, for corresponding values of x co-ordinates.

Domain & Range
The set of x co-ordinates is called as the domain and the set of y co-ordinates is called as the Range of Function.

Types of Functions

 * 1) One-One : If each element of Domain points to exactly one  element in Range.
 * 2) Many-One : If many elements of Domain are pointing to a single element in Range.
 * 3) Into   : Some elements of Range remain unpointed by Domain.
 * 4) Onto  : Some elements of Domain do not point to Range.

Inverse Function
If y = f(x)

then f-1(y) = x

Composite Function
A function defined within a function is called as a Composite function.

e.g. g(f(x))

thus, the domain of f is x , while domain of g is f(x).

Intervals
Functions can also b defined within certain intervals. Many times, periodic functions like sin and cosine are defined using intervals.

Tips and Tricks
1) For functions, try to plot graphs . So learn more different kinds of functions and their graphs , or else derive the graph using x-y method.

2) For Set theory draw venn diagrams for better understanding of the question.