Relations and Functions Class 12 Notes Mathematics Chapter 1

CHAPTER AT A GLANCE

  • Ordered pair – A pair of objects (elements) taken (listed) in a definite (specific) order is known as an ordered pair. In the ordered pair (ai, bi), ai is known as first element and bi is known as second element.
  • Equality of ordered pairs – Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2
  • Cartesian product – Let A and B be two non-empty sets, then the set of all distinct ordered pair of the form (ai, bi), where a1 ÎA and b1 ÎB is called
    cartesian product of A and B and is denoted by A × B. If A and B have m and
    n elements respectively, then, number of order pairs in A × B is equal to m × n.
  • Relations – Let A and B be two non-empty sets then every subset of A × B
    defines a relation from A to B and every relation from A to B is a subset of A × B.
  • Domain (D) and Range (R) of a Relation – If R be a relation from set A to B
    then the set of first entries of all ordered pairs in R is called domain and set of
    all second entries in all order pairs of R is called range of R.
  • Inverse Relation – Let A and B be two sets and let R be a relation from set A to
    set B then the inverse of R, denoted by R–1, is a relation from set B to set A
    and is defined as.
  • Number of Relation – Let A and B be two non-empty finite sets having m and
    n elements respectively then the number of relations from A to B is (2)m×n
  • Universal Relation – A Relation R in a set A is called a universal relation if R = A × A
  • Empty Relation: A relation R in a set A is called an empty relation, if no element
    of A is related to any element of A.
  • Equivalence Relation – Any relation on a set A which is Reflexive, Symmetric and Transitive is called an equivalence Relation.
  • Function: A function from a non-empty set A to a non-empty set B is a relation
    from set A to set B such that each element of set A is associated with one and
    only one element (i.e. unique element) of set B (i.e. each element of set A is
    associated with a unique element of set B)
    Note that more than one element of A may be associated with any element of
    B but in set B, there may be one or more elements to which no element of set A
    is associated.
  • Domain – The set A is called the domain of ‘f’ (denoted by Df )
  • Co-domain – The set B is called the co-domain of ‘f ’ (denoted by Cf)
  • Range – The range of ‘f ’ denoted by Rf is the set consisting of all the image of the elements of the domain A.
  • One-One function (Injective) – Let f : A ® B then f is called a one-one function
    if no two different elements in A have the same image i.e. different elements in
    An associated with different elements in B.
  • Onto function (Surjective) : A function f : A ® B is said to be onto function,
    if every element of B is the image of some element of A under f.
    Note : f : A ® B is onto iff Range of f = B.
  • One-One Onto function (Bijective) : A function f : A –> B is said to be one-
    one and onto if f is both one-one and onto.
  • Many-One – A function f : A –> B is said to be many-one if two or more elements
    of set A have the same image in B.
  • Even and Odd function – A function f(x) is said to be even function iff
    f(–x) = f(x) and A function f(x) is said to be odd function iff f (–x) = – f(x).
  • Composition functions – Let f : A –> B and g : B –> C be two functions. Then the composition of f and g, denoted by gof : A –> C given by
  • Invertible Function and Inverse Function: A function f : X -> Y is defined to be invertible if there exists a function g : Y ® X such that g of = Ix and fog = Iy . The function g is called the inverse of f and is denoted by f–1 . Thus, f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

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