Note on Relation

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Generally, relation refers to the blood connection between the people. For example, brother and sister, father and son etc.
But in mathematics, if A and B are two sets in which A × B is called the cartesian product of A and B, any subset of A × B is called a relation from A to B. The relation from A to B is denoted by R. The relation from A is called a relation on A.
For examples,
Let, A = {3, 5} and B = {4, 6}
Then, A × B = {3, 5} × {4, 6}
= {(3, 4), (3, 6), (5, 4), (5, 6)}
Let, R = {(3, 4), (5, 6)}
So, R is a subset of A × B.
So, R is a relation from A to B.

Representation of a relation

A relation can be represented in 5 ways as:

  • By the set of ordered pairs
  • By standard description using a rule of formula
  • By table
  • By graph
  • By arrow diagram

By the set of ordered pairs
A relation can be represented in the set of ordered pairs in which there are two components. The first component is called x-component and the second component is called y-component.
For example,

  • R = {(Sulav, Shree), (Siddartha, Bhagwat), (Asmita, Riya)}
    R = {(1, 1), (2, 2), (3, 3)}

By standard description using a rule or a formula
A relation can be represented by using the formulas and standard description.
For example,

  • Let, A = {3, 4, 5} and B = {3, 4, 5}
    Then,
    A × B = {(3, 3), (3, 4), (3, 5), (4, 3), (4, 4), (4, 5), (5, 3), (5, 4), (5, 5)}
    Here, R is a subset of A × B. So, R is a relation from A to B. The relation R can be shown as, R = {(x, y) : x=y}

  • Let, A = (5, 6) and B = (7, 8)
    Then,
    A × B = (5, 6) × (7, 8)
    = {(5, 7), (5, 8), (6, 7), (6, 8)}
    Let, R = {(5, 7), (5, 8), (6, 8)}
    Here,R is a subset of A × B. So R is a relation from A to B. The relationship R can be represented as,R = {(x, y) : x<y}

By Table
A relation can also be represented in table which one as follows: -

  • Let, A = {2, 4} and B = {4, 6}
    Then, A × B = {2, 4} × {4, 6}
    = {(2, 4), (2, 6), (4, 2), (4, 6)}
    Then, R is a relation from A to B. This relation can be represented as,
    x 2 4
    y 4 6

By Graph
A relation can also be shown in graphs by the use of graph paper. For example,

  • Let, A = {4, 5, 6} and {4, 5, 6}
    Then, A × B = {4, 5, 6} × {4, 5, 6}
    = {(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}
    So, R = {(4, 4), (5, 5), (6, 6)}
    Here, R is a relation from A to B which is represented as,
    Example of graph

Example for graph

By Arrows Diagram

  • Let, A = {2, 4} and B = {4, 6}
    Then A × B = {2, 4}×{4, 6}
    = {(2, 4), (2, 6), (4, 4), (4, 6)}
    Let, R = {(2, 4), (4, 6)}
    So, R is a relation from A to B. This relation can be represented as,
    .

Example for arrow diagram

Domain and Range of a Relation

Let R be a relation from A to B. Then, the set of all first elements of the ordered pairs of R are called the domain. For example,
Let, A = (1, 2) and B = (3, 4)
Domain of relation = (1, 3)
Similarly, the set of all the second elements of the ordered pair of R is called the range. For example,
Let, A = (1, 2) and B = (3, 4)
Range of a Relation = (2, 4)

Inverse relation

The interchanging of x-component and y-components of each pair of the relation of R is denoted by R-1.
For example,
Let, A = {1, 2} and B = {3, 4}
Here, A × B = {1, 2} × {3, 4} = {(1, 3) (1, 4) (2, 3) (2, 4)}
Thus, R-1= {(3, 1), (4, 1), (3, 2), (4, 2)}

  • A relation between two sets is a collection of ordered pairs containing one object from each set.
  • Any subset of A × B is called a relation from A to B. 
  • The relation is denoted by R.
  • The inverse relation is denoted by R-1.
.

Very Short Questions

Solution:

Here,
R = {(2, 4), (3, 6), (4, 8), (5, 10)}

∴ R−1 = {(4, 2), (6, 3), (8, 4), (10, 5)}

Solution:

Here,
R = {(a, w), (b, x), (c, y), (d, z)}

∴ R−1 = {(w, a), (x, b), (y, c), (z, d)}

Solution:

Here,
R = {(1, 3), (5, 7), (9, 11), (13, 15)}

∴ R−1 = {(3, 1), (7, 5), (11, 9), (15, 13)}

Solution:

Here,
R = {(p, a), (q, b), (r, c), (s, d)}

∴ R−1= {(a, p), (b, q), (c, r), (d, s)}

Solution:

Here,
R = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 4)}
Domain of R = a set of x-components of each ordered pair of the relation R.
= {1, 1, 2, 2, 3}
= {1, 2, ,3}

Range of R = a set of y-components of each ordered piar of the relation R.
= {2, 4, 2, 4, 4}
= {2, 4}

Solution:

Here,
Domain of R = set of x-component
= {2, 2, 2, 3, 3}
= {2, 3}

Range of R = set of y-component
= {a, b, c, a, b}
= {a, b, c}

Solution:

Here,
Domain of R = set of x-component
= {−1, 2,−3, 4}

range of R = set of y-component
= {2,−4, 6,−8}

Solution:

Here,
Domain of R = {(a, b, c, a, b}
= {a, b, c}

Range of R = {p, q, r, q, r}
= {p. q. r}

0%
  • Find the inverse of the given relation.
    {(2, 4), (3, 6), (4, 8), (5, 10)}

    R−1 = {(4, 2), (6, 3), (8, 4), (10, 5)}
    R−1 = {(10, 5), (4, 2), (6, 3), (8, 4) }
    R−1 = { (6, 3), (4, 2),(8, 4), (10, 5)}
    R−1 = {(4, 2), , (8, 4), (6, 3)(10, 5)}
  • Find the domain of given relation.
    {(a, p), (b, q), (c, r), (a, q), (b, r)}

    Domain of R = {(b, a, b, c, a, }= {a, b, c}
    Domain of R = {(a, b, c, a, b}= {a, b, c}
    Domain of R = {(a, b, a, c,  b}= {a,  c, b}
    Domain of R = {(a,c, b, a, b}= {a, b, c}
  • Find the  range of given relation.
    {(a, p), (b, q), (c, r), (a, q), (b, r)}

    Range of R = {p, q, r, q, r}= { q, p, r}
    Range of R = {p, r, q, q, r}= {p, q, r}
    Range of R = {p, q, r, q, r}= {p, q, r}
    Range of R = {p, r, q,  q, r}= {p, q, r}
  • Find the inverse of the given relation.
    {(a, w), (b, x), (c, y), (d, z)}

    R−1 = {(w, a), (x, b), (y, c), (z, d)}
    R−1 = {(w, a), (y, c), (x, d), (z, d)}
    R−1 = {(w, a), (y, c), (x, d),  (z, d)}
    R−1 = {(x, d), (w, a), (y, c), (z, d)}
  • Find the inverse of the given relation.
    {(1, 3), (5, 7), (9, 11), (13, 15)}

    R−1 = {(11, 9), (3, 1), (7, 5), (15, 13)}
    R−1 = {(3, 1), (7, 5), (11, 9), (15, 13)}
    R−1 = { (7, 5), (3, 1), (11, 9), (15, 13)}
    R−1 = {(3, 1), (11, 9), (7, 5), (15, 13)}
  • Find the domain of the given relation.
    {(2, a), (2, b), (2, c), (3, a), (3, b)}

    {2, 3}
    {2, a}
    {3, 3}
    {2, 2}
  • Find the range of the given relation.
    {(2, a), (2, b), (2, c), (3, a), (3, b)}

    {a, 3, c}
    {c, b,a}
    {a, b, c}
    {2, b, c}
  • Find the domain of given relation.
    {(−1, 2), (2, −4), (−3, 6), (4, −8)}

    Domain of R =  {3, 2,1, 4}
    Domain of R = {−4,3, 2,- 1}
    Domain of R = {2, 1,−3, 4}
    Domain of R =  {−1, 2,−3, 4}
  • Find the range of given relation.
    {(−1, 2), (2, −4), (−3, 6), (4, −8)}

    range of R = {2, 6, 8,−4}
    range of R = { 8, 4, 6, 2}
    range of R = {2,−4, 6,−8}
    range of R = {-8, 6, -4, 2}
  • Find the domain of given relation.

    (1, 2),  (1, 4), (2, 2), (2, 4), (3, 4)

    1, 2, 3
    1, 2, 2
    1, 2, 3, 4
    4, 3, 2, 1
  • Find the range of given relation.

    (1, 2),  (1, 4), (2, 2), (2, 4), (3, 4)

    1, 3, 2
    4, 2, 1
    2, 4
    2, 2
  • Find the domain of given relation.

    (2, a),  (2, b), (2, c), (3, a), (3, b)

    2, a, 3, b
    2, 3
    a, b, c
    a, b, c, 2, 3
  • Find the range of given relation.

    (2, a),  (2, b), (2, c), (3, a), (3, b)

    a, b, c, 2, 3
    2a, 2b, c
    a, b, c
    2, 3
  • Find the inverse of given relation.

    (2, 4),  (3, 6), (4, 8), (5, 10)

    (10, 5),  (8,4), (3, 6), (2, 4)
    (2, 4),  (3, 6), (4, 8), (5, 10)
    (4, 2),  (6, 3), (8, 4), (10, 5)
    (5, 10), (4, 8), (3, 6), (2, 4) 
  • Find the inverse of given relation.

    (1, 3), (5, 7), (9, 11), (13, 15)

     

    (3, 1), (7, 5), (11, 9), (15, 13)
    (15, 13), (11, 9), (7, 5), (3, 1)
    (1, 3), (5, 7), (9, 11), (13, 15)
    (13, 15), (9, 11) , (5, 7), (1, 3)
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