Coupon Accepted Successfully!


Relation on a Set

Consider two sets A and B. Every subset of the product set A × B is called a relation between A and B. It is denoted by R.

Hence, we can say that Description: 56596.png.

If (x, y)  R, then we can write it as x R y and is read as ‘x is related to y’.

If A = B, then R is a relation on A.

The set of first elements of the ordered pair that belongs to R is known as the domain of the relation R. It is denoted by D.

D = {x: (xy R, for x  Ai.e.Description: 56616.png

The set of all second elements of the ordered pair that belongs to R is known as the range of the relation R. It is denoted by E.

E = {y: (xy R, for y  Bi.e.Description: 56606.png

Let R be a relation on the set of whole numbers W, defined by a + 4b = 24. Find
(i) R
(ii) Domain of R
(iii) Range of R
We have R = {(ab) : a ∈ Wb ∈ Wa + 4b = 24} = {(24, 0), (20, 1), (16, 2), (12, 3), (8, 4), (4, 5), (0, 6)}
The above set of ordered pairs is obtained by taking b = 0, 1, 2, 3, 4, 5, 6 in the given relation and finding the value of a. If b > 6, then the value of ‘a’ obtained becomes negative which does not belong to the set W.
Domain of R = {24, 20, 16, 12, 8, 4, 0}
Range of R = {0, 1, 2, 3, 4, 5, 6}

Different Types of Relation

  • Identity relation: A relation R on a set is said to be an identity relation, if both the elements of the ordered pairs are same.
    The identity relation on a set A is shown as IA = {(xx) : x  A}

    Example: Let A = {1, 2, 3}, then IA = {(1, 1), (2, 2), (3, 3)}

  • Universal relation: A × A is called the universal relation on a set A.

    Example: Let A = {2, 4, 6}, then R = A × A = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}

  • ƒ Void relation: If a relation R from A to B is a null set, then R is said to be a void relation.

    Example: Let A = {1, 3} and B = {9, 13}.

    Let R = {(m, n) : m  An  Bm × n is even}.

    Since none of the numbers (1 × 9), (1 × 13), (3 × 9), (3 × 13) is even, R is an empty set.

  • ƒ Reflexive relation: The relation R on a set ‘A’ is said to be a reflexive relation, if R contains all possible ordered pairs of the type (xx) for all x  A.

    Example: Let A = {3, 6, 9}

    The relation R = {(3, 3), (3, 6), (6, 6), (9, 9), (9, 6)} is a reflexive relation, since all possible pairs of (xx) type are present in R.

    The relation R = {(3, 3), (3, 6), (6, 9), (9, 9), (9, 6)} is a not reflexive relation, since (6, 6) is missing in R.

Note: R is a reflexive relation, if every element in a set ‘A’ is related to itself.

  • Symmetric relation: The relation R on a set A is said to be symmetric, if (xy R and yx R.
    It means that, for each ordered pair (xy) if the reverse pair ( yx) is also there in R, then R will be symmetric.

    Example: Let S be a set of all straight lines in a plane. Then the relation R in S defined by, ‘a is perpendicular to b’. Then the relation R is said to be symmetric, since a straight line ‘a’ is perpendicular to another line ‘b’, and the line ‘b’ is also perpendicular to line ‘a’.

  • ƒ Transitive relation: The relation R on a set A is said to be transitive if (xy R and ( yz R  ( xz R

    Example: Let there be a set of straight lines in a plane. Then the relation R on the set of lines defined by ‘a is parallel to b’ is a transitive relation, since if a line a is parallel to a line b and the line b is parallel to another line c, then the line a is parallel to c.

  • The relation ‘a is perpendicular to b’ is not a transitive relation. This is because if a line a is perpendicular to a line b and if the line b is perpendicular to a line c, then a and c will be parallel to each other.
  • Equivalence relation: The relation R on a set A is said to be an equivalence relation, if R is reflexive, symmetric and transitive.

    Example: The relation R = {(xy) : x  Ay  Ax = y} is an equivalence relation.

    We can see that,
    x = xi.e., reflexive
    x = y  y = xi.e., symmetric

    x = yy = z  x = zi.e., transitive

  • ƒ Inverse relation: If R is a relation from the set A to B, then the inverse relation R-1 from B to A is defined by R-1 = {(ba) : (ab R}

    Example: Let A = {1, 2}, B = {a, b} and R = {(1, a), (2, a), (1, b), (2, b)} be a relation from A to B.


    Then inverse relation of R is R -1 = {(a, 1), (a, 2), (b, 1), (b, 2)}.


Let A and B be two non-empty sets. Then, if each element x of A is related with a unique element f (x) of B, it is called function from A to B and it is denoted by f A  B.


The element f (x) is called the image of x, while x is called the pre-image of f (x).


Let f A  B, then A is called the domain of f, while B is called the co-domain of f.


The set f (A) = { f (x) : x  A} is called the range of f.


Example : Let A = {1, 2, 3, 4}, B = {4, 9, 16, 25}
Let f (x) = x2
Then, f is not a function, since no element of B is assigned to the element 1 ∈ A.


Example: Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10, 12, 14}
Let f(x) = 2x
Then f(1) = 2
f(2) = 4
f(3) = 6
f(4) = 8
f(5) = 10
We can see that each element of A has a unique image in B.
So f : A → B is a function from A to B.

Various Types of Functions

  • One–one function: If distinct elements in A have distinct images in Bi.e.f is ‘one-one’ if f(x1) = f(x2 x1 = x2. It is also called as injective function.

    Example: Let fA  B be defined by f(x) = x2.

    We know that, 1 ≠ -1, but f(1) = f(-1). So ‘f ’ is not one–one.

    Let fA  B be defined by f(x) = x3.


    Here, f is one–one as f(x1) = f(x2 x1 = x2.


  • ƒMany–one function: If two or more than two elements of set A have the same image in B, then it is called ‘many–one’ function.

    Example: Let fA  B be defined by f(x) = x2.

    We know f(-1) = (-1)2 = 1 and also f(1) = (1)2 = 1.
    So two elements of set A have the same image in set B. Hence ‘f’ is a many—one function.


  • Onto function: A function f : A → B is called onto or surjective if f(A) = B, i.e., for all b ∈ B there at least one a ∈ A with f(a) = b
    It is also called as surjective function. In order words, no element in B should be without a pre-image. For onto functions, range = co-domain.

    Example: Let A = {1, 2, 3}, B = {a, b} and let f = {(1, a), (2, a), (3, b)}.


    Here, the domain is A and no two components of f have the same first element. Hence it is a function. Also to every element y in B, there is an element in A and hence f is an onto function.


  • Into function: If at least one element in B has no pre-image in A, it is called into function.

    Example: Let A = {1, 2, 3}, B = {a, b, c, d } and let f = {(1, a), (2, b), (3, c)}

    It is clear that f is a function from A to B.
    Here the element d in B does not have a pre-image in A.

    Hence f is an into function.


  • Bijective function: A one–one and onto function is said to be bijective function.
  • Equal functions: If two functions f and g have the same domain and satisfy the condition f(x) = g(x), then f and g are said to be equal functions.
  • Composite function: Let f A  B and gB  C be two functions. The function from A to C which maps an element x  A into gf (x))  C is called a composite function of ‘f ’ and ‘g’. It is represented as ‘g o f ’.

    Example: Let A = {1, 2, 3}, B = {w, x, y, z}, C = {2, 4, 6, 8, 10}

     f = {(1, w), (2, x), (3, y)} and g = {(w, 2), (x, 4), (y, 6), (z, 8)}

    Then gof is the function {(1, 2), (2, 4), (3, 6)}


  • Inverse function: Let f be a bijective function from A to B and f (x) = y.
    Then we can define a function f -1 such that f -1y) = x.
    The above function f -1y) is called the inverse of f.

Note: A function has an inverse function if and only if it is bijective.


If f(x) = 2 + x, then f -1(x) = ?
Let y = f(x)
Description: 59089.png 

Test Your Skills Now!
Take a Quiz now
Reviewer Name