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Sets, Relations and Functions

Sets, Relations and Functions PDF Notes, Important Questions and Synopsis

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 (begin mathsize 12px style straight A intersection straight B intersection straight C intersection. ... end style ).
    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 (begin mathsize 12px style straight A union straight B union straight C union. ... end style ).
    iv. begin mathsize 12px style text A∪B∪C=A+B+C−(A∩B)−(B∩C)−(C∩A)+(A∩B∩C end text right parenthesis end style
    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)  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. begin mathsize 12px style open parentheses straight f over straight g close parentheses left parenthesis straight X right parenthesis equals fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight g left parenthesis straight x right parenthesis end fraction end style, where g(x)≠0
    Note: Domain for all the above functions is begin mathsize 12px style straight A intersection straight B. end style

  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, " 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

     begin mathsize 12px style straight real numbers minus open curly brackets straight pi over 2 plus nπ comma straight n element of straight natural numbers close curly brackets end style

    (-∞, +∞)

    f(x) = cosec x

     Error converting from MathML to accessible text.

    (-∞,-1] U [1, +∞)

    f(x) = sec x

     Error converting from MathML to accessible text.

    (-∞,-1] U [1, +∞)

    f(x) = cot x

     Error converting from MathML to accessible text.

    (-∞, +∞)





  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

    begin mathsize 12px style open square brackets negative straight pi over 2 comma straight pi over 2 close square brackets end style

    [0,Π]

    begin mathsize 12px style open square brackets negative straight pi over 2 comma straight pi over 2 close square brackets end style

    (0,Π)

    begin mathsize 12px style open square brackets open 0 comma straight pi over 2 close parentheses close union open parentheses open straight pi over 2 comma straight pi close square brackets close end style begin mathsize 12px style open square brackets open negative straight pi over 2 comma 0 close parentheses close union open parentheses open 0 comma straight pi over 2 close square brackets close end style


                

  3. Exponential functions

           

  4. Logarithmic functions

  5.  Absolute value function
    |x|=begin mathsize 12px style open curly brackets table row cell negative straight x semicolon text  x<0 end text end cell row cell text end text straight x semicolon text  x end text greater or equal than text 0 end text end cell end table close end style

  6. Greatest integer function
    y=[x]=begin mathsize 12px style open curly brackets table row cell text end text straight x semicolon text x end text element of text I end text end cell row cell table attributes columnalign left end attributes row cell text Greatest integer  end text end cell row cell text less than x end text semicolon text end text otherwise end cell end table end cell end table close end style

  7. Signum function
    begin mathsize 12px style straight y equals sgn left parenthesis straight x right parenthesis equals open curly brackets table row cell text end text 1 semicolon text end text straight x greater than 0 end cell row cell text end text 0 semicolon text  x=0 end text end cell row cell negative 1 semicolon text  x<0 end text end cell end table close end style

  8. Fractional part function
    begin mathsize 12px style straight y equals straight f left parenthesis straight x )={ straight x }= straight x −[ straight x right square bracket end style

  9. Even function
    begin mathsize 12px style straight f left parenthesis negative straight x right parenthesis equals straight f left parenthesis straight x right parenthesis end style i.e. symmetric about the y-axis

  10. Odd function
    begin mathsize 12px style straight f left parenthesis negative straight x right parenthesis equals negative straight f left parenthesis straight x right parenthesis end style i.e. symmetric about the origin

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