Sets, Relations and Functions

SYNOPSIS

1. Set:
   A set is a well-defined collection of objects and it is denoted by capital letters A, B, ..., Z.
2. Representation of a set: A set can be represented in two forms:
   i. Roster form: All the elements are listed and separated by commas inside the { } braces.
   ii. Set builder form: It is the mathematical representation of a set where all the members share a common property listed in the { } braces.
3. Cardinality of a set: Number of elements present in a set.
   
   
   
4. Types of sets:
   
   i. Empty set: A set having no element or cardinality zero is an empty set, i.e. { } or Ø. 
   ii. Finite set: A set having finite number of elements is a finite set, e.g. A = {1, 2, 3}.
   iii. Infinite set: A set having infinite number of elements is an infinite set, e.g. B = {1, 2, 3,...}.
   iv. Equal sets: Two sets are equal when they share the same elements and have equal cardinality.
   v. Subset: A is a subset of B if B contains all the elements of A. 
   Note: If A is a subset of B and not equal to B, then A becomes the proper subset of B.
   vi. Superset: If B contains all the elements of A, then B is the superset of A.
   vii. Power set: The power set of any set A is a set of all the subsets of A and it is denoted by P(A).

5.  Operations on sets:

   i. Disjoint sets: When the sets have no common element, they are called disjoint sets.
   ii. Intersection of sets: Intersection of two/more sets is a part (set of elements) which is/are common in those sets ( ).
   iii. Union of sets: Union of two sets A and B is a set containing all the elements of A as well as B.
   Same applies for n number of sets ( ).
   iv.
   v. Complement of a set: Complement of set A is a set containing all the elements not in A and denoted by A’.
   vi. Difference of two sets: Difference of two sets A and B (A⧍B) is a set containing all the elements of A and B which are not common.
   vii. Cartesian product of sets: It is a set of ordered pair of elements containing one object from each set.
   It is denoted by A × B, where the first object belongs to the first set and the second object belongs to the second set.

Relations

1. Relation: A relation R between two sets is a collection of ordered pairs containing one object from each set.
   It can also be written as a Cartesian product of two sets, i.e. R = A × B, where all the elements share a common property.

2.  Types of relations:
   i. Reflexive: A relation R is reflexive if ∀ x ∊ X, (x, x) ∊ R.
   ii. Symmetric: A relation R is symmetric if (a, b) ∊ R implies (b, a) ∊ R.
   iii. Transitive: A relation R is transitive if (a, b) ∊ R and (b, c) ∊ R implies (a, c) ∊ R.
   Note: If R is reflexive, symmetric and transitive, then it is an equivalence relation.
   iv. Identity: A relation R is an identity if R = {(x, x):x ∊ X}.

Function:

1. Function:A function is a relation where each input has a single output.
   It is written as f(x), where x is the input.
2. Domain, Co-domain and Range of a function:
   Let f be a function from set A to set B, i.e. f: A→B, then A is the domain and B is the co-domain of f.
   Here, all the inputs belong to A and the outputs belong to B.
   Set containing all the outputs is the range of a function which is denoted by f(A) = {f(a): a ϵ A}.
   Note: Clearly, f(A) ϵ B.
3. Real-valued function:
   A function with domain and range both being subsets of a set of real numbers.

4. Operations on functions:
   Let f, g: A→B be two real-valued functions, then
   i. (f ± g)(x) = f(x)± g(x)
   ii. (f • g)(x) = f(x) • g(x)
   iii. , where g(x)≠0
   Note: Domain for all the above functions is 

5.  Classification of functions:

1. One–one function

   A function is a one–one function when each element of domain A is connected with a different element of co-domain B. It is also called injective function.
   i.e. For a function f: A→B if "x, y ∊ A such that f(x) = f(y) ⇒ x = y
   
2. Many–one function
   When any two or more elements of domain A are connected with a single element of co-domain B, then the function is said to be a many–one function.
   
3. Onto function
   A function f is said to be an onto function if each element of co-domain B is connected with the elements of domain A, i.e. if the range is the same as the co-domain.
   It is also called a surjective function.
   i.e. f: A→B, "y ∊ B, ∃ x ∊ A such that y = f(x).
   
   
   
4. Into function
   A function f is said to be an into function if co-domain B has at least one element which is not connected with any of the elements of domain A.
   
   
   
   
5. Bijective function
   A function f is said to be a bijective function if it is one–one and onto.

1. Types of functions:
   Trigonometric functions
   

   Function	Domain	Range
   f(x) = sin x	(-∞, +∞)	[−1,1]
   f(x) = cos x	(-∞, +∞)	[−1,1]
   f(x) = tan x		(-∞, +∞)
   f(x) = cosec x		(-∞,-1] U [1, +∞)
   f(x) = sec x		(-∞,-1] U [1, +∞)
   f(x) = cot x		(-∞, +∞)

   
   
   
   
   
2. Inverse trigonometric functions
   
   

   	sin-1x	cos-1x	tan-1x	cot-1x	sec-1x	cosec-1x
   Domain	[−1,1]	[−1,1]	(−∞,∞)	(−∞,∞)	(−∞,−1]U[1, ∞)	(−∞,−1]U[1, ∞)
   Range		[0,Π]		(0,Π)

   
               

3. Exponential functions
   
          

4. Logarithmic functions
   
   

5.  Absolute value function
   |x|=
   
   

6. Greatest integer function
   y=[x]=
   
   

7. Signum function
   
   
   

8. Fractional part function
   
   
   

9. Even function
    i.e. symmetric about the y-axis
   
   

10. Odd function
     i.e. symmetric about the origin