The Relations

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In our daily life, we come across many relations such as Mother–daughter, Coach–player, teacher–student and many more.

In mathematics too, we come across relations such as A is a subset of B, line l is perpendicular to line m, number x is greater than number y. In all these, we notice that a relation involves pairs of objects in certain order.

A "relation" is just a relationship between sets of information. Think of all the students in your class and their marks in Maths slip test. The pairing of names and marks is a relation. In relations, the pairs of names and marks are "ordered", which means one comes first and the other comes second i.e., we could set up this pairing so that either you give me a name, and then I give you that student’s marks, or else you give me the marks, and I give you the names of all the students who scored those marks. Hence we can say that a relation is simply a set of ordered pairs.

An ordered pairis a pair of entries in the specific order. The two entries are separated by comma and enclosed within brackets.

The ordered pair (a, b) contains two elements ‘a’ and ‘b’. The entry ‘a’ is called the first element or first component or first coordinate and ‘b’ is called the second element or second component or second coordinate.

By interchanging the positions of the components, the ordered pair is changed.

                                                   Thus, (5,9) ( 9,5)

Example: In geometry, the position of a point in a plane is determined by an ordered pair. As shown in the figure, the ordered pairs (3, 1) and (1, 3) represent two different points A and B respectively.

Equality of two ordered pairs: Two ordered pairs are said to be equal only when their first as well as second components are equal. Two ordered pairs (a, b) and (c, d) are equal, if a = c and b = d.

Cartesian Product of Two Sets

Let A and B are two non–empty sets. Then, the cartesian product of A and B, written as A × B(read as A cross B) is the set of all ordered pairs (a, b) such that a A and b B.

i.e., A × B = { (a, b) / a A and b B}.

Note:

1. A × B ≠ B × A
2. A × B = B × A   is true only when A = B
3. If n(A) = x and n(B) = y then n(A × B) = x * y = xy
4. If A = { } and B = { } then A × B = { }

Representation of cartesian product of two sets: The cartesian product of two sets can be represented in three ways:

1. Arrow diagram
2. Tree diagram
3. Graphical representation

Arrow Diagram: We may represent the cartesian product of the sets by an arrow diagram.

Example 1: If A = {3, 4, 5} and B = {6, 8, 10} then represent A × B in an arrow diagram?

Sol:   Given A = {3, 4, 5} and B = {6, 8, 10}

          Here n(A) = 3   and   n(B) = 3

          Then   n(A × B)  =  3 * 3  =  9

             A × B = {3, 4, 5} × {6, 8, 10}

          Now the arrow diagram for A × B is as follows:

From the above diagram we can conclude the Ordered pairs as:

          A × B = {(3, 6), (3, 8), (3, 10), (4, 6), (4, 8), (4, 10), (5, 6), (5, 8), (5, 10)}

              n(A × B) = 9

Example 2: If A = {p, q} and B = {k} then represent A × B as an arrow diagram?

Sol:  Given A = {p, q} , B = {k}

              n(A) = 2 , n(B) = 1

              n(A × B) = 2 * 1 = 2

          Now the Arrow diagram for A × B is as follows:

          From the above diagram we can conclude the set of ordered pairs as:

              A × B = { p, q} × {K} = { (p, k), (q, k)}

Tree Diagram:We may represent the cartesian product of the sets by a tree diagram.

Example 1: Let P = {a, b, c} and Q = {m, n} be two sets. Represent P × Q in a tree diagram?

Sol: Given P = {a, b, c} and Q = {m, n}

              n(P) = 3   ,   n(Q) = 2

           n(P × Q) = 3 * 2 = 6.

        Now the Tree diagram for P × Q is as follows:

        From the above diagram:

        P × Q = { a, b, c} × {m, n}

            P × Q = { (a, m), (b, m), (c, m), (a, n), (b, n), (c, n) }.

Example 2:   From the following Tree Diagram find A × B?

Sol:

        From above Diagram we can conclude that

        A = { 3, 5} and B = { 9, 11, 13, 17}

            n(A) = 2 and n(B) = 4

            n( A × B) = 2 * 4 = 8.

            A × B = {3, 5} × { 9, 11, 13, 17}

             A × B = {(3, 9), (3, 11), (3, 13), (3, 17), (5,9), (5, 11), (5, 13), (5, 17)}.

Graphical representation of cartesian product: The cartesian product can be represented in a graphical form.

Example: If A = {1, 2, 3, 4} and B = {1, 2, 3} then represent A × B in a Graph?

Procedure:

Draw two lines one horizontal line and vertical line which are perpendicular to each other. Represent the first set elements 1, 2, 3, 4 on horizontal line and represent the second set elements 1, 2, 3 on vertical Line.

Draw lines 1, 2, 3, which are parallel to the horizontal line.

Similarly draw lines from 1, 2, 3, 4 which are parallel to vertical line.

The intersection of vertical line and horizontal line are represented as ‘* ’.

The * represents the ordered pairs of the set A × B.

So, from the graph the ordered pairs are:

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

    The set A × B =

{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3)}

Math Relations

Some times it becomes necessary to study various types of relations in different branches of science. Particularly in Mathematics we are familiar with the connection between a point and a line, a line and a plane , a plane and a space.

In every day life we come across various relations such as:

1. Mary is the wife of John
2. a is equal to b
3. x is greater than y
4. Line L is parallel to line M
5. Line P is perpendicular to line Q
6. A is subset of B

Sometimes we use symbols like > , < , ≠ ,   to denote specific relations.

In general, we use the letter R to denote a relation between two elements.

Example:

If a and b are in relation R we write a R b or (a, b) R.

Example:

If R stands for the relation "is equal to" then a R b implies a is equal to b and b R a means b is equal to a. In this context both a R b and b R a are true.

Example:

If R is the relation "is greater than" , 5 R 3 means 5 > 3, which is true. But 3 R 5 implies 3 > 5, which is not true.

Note: Given a relation R, a R b need not imply b R a. That is a R b and b R a can be distinct. That is, (a, b) R need not imply (b, a) R

Another definition for Relation:

We know that every relation is a set of ordered pairs and the cartesian product of two sets is a set consists of all ordered pairs.

      A relation can be defined as a subset of the cartesian product of two sets.

Example: If A = {1, 2, 3} and B = {1, 2} then find A × B ?

Sol:     Given A = { 1, 2, 3} and B = {1, 2}.

                A × B = {(1, 1), (1, 2), (2, 1), (2, 2), ( 3, 1), (3, 2)}.

            From A × B we can write the following relations that is:

            R1 = {(1, 1), (2, 2)} Since R1 consists of ordered pairs of A × B.

                R1 A × B.

            Here the coordinates of ordered pair are equal.

            We can say that the relation is "equal to".

            R2 = {(1, 2)} Since R2 consists of ordered pairs of A × B .

                R2 A × B.

            Here first coordinate is less than the second coordinate.

            We can say that the relation is " is less than" .

            R3 = {(3, 1), (3, 2), (2, 1)} Since R3 consists of ordered pairs of A × B.

                R3 A × B

            Here first coordinate is greater than the second coordinate.

            We can say that the relation is "is greater than".

Domain and Range of a relation:

The set of first coordinates of all the ordered pairs of a relation R is called the Domain of R, while the set of second coordinates of all the ordered pairs of R is called the Range of R.

Example 1: Find the domain and range of the relation R, where R =

{(1, 2), (1, 3), (2, 2), (2, 3), (4, 2), (4, 3), (5, 2), (5, 3), (6, 2), (6, 3)} ?

Sol:     Given Relation R =

                      {(1, 2), (1, 3), (2, 2), (2, 3), (4, 2), (4, 3), (5, 2), (5, 3), (6, 2), (6, 3)}.

            Domain and range for the above relation is:

            Domain = First co–ordinates of the ordered pairs ={1, 2, 4, 5, 6}

            Range = Second co–ordinates of the ordered pairs = {2, 3}.

Example 2: Find the domain and range of the relation R, where R ={(a, a), (a, b)}?

Sol:     Given Relation R = { (a, a), (a, b)}

            Domain and range for the above relation is:

            Domain = First co – ordinates of the ordered pairs = {a} and

            Range = Second co – ordinates of the ordered pairs = {a, b}.

Rule of the relation or formula of the relation:

The property that tells us how the first coordinate is related to the second coordinate of each ordered pair in a relation R is called the rule of the relation.

Example 1: If A = {10, 11, 12, 13, 14} and B = {13, 14} find A × B, also find the formula for the following relations:

1) R1 = {(13, 13) ,(14, 14)}

2) R2 = { (14, 13)}

3) R3 = {(10, 13), (10, 14), (11, 13), (11, 14), (12, 13), (12, 14), (13, 14)} ?

Sol:     Given sets are A = {10, 11, 12, 13, 14} and B = {13, 14}

            Then A × B =

                  {(10, 13), (10, 14), (11, 13), (11, 14), (12, 13), (12, 14), (13, 13),

                  (13, 14), (14, 13), (14, 14)}

            Given relations are R1 , R2 , R3 .

Formulas for the given relations:

1. R1 = {(13, 13) ,(14, 14)}
   Here R1 consists of ordered pairs of A × B, whose first and second coorrdinates are equal.
   Hence the formula for the Relation R1 is ‘ Is Equal To ’.
2. R2 = { (14, 13)}
   Here R2 consists of ordered pairs of A × B, whose first coorrdinate is greater than the Second coordinate.
   Hence the formula for the Relation R2 is ‘ Is Greater Than ’.
3. R3 = {(10, 13), (10, 14), (11, 13), (11, 14), (12, 13), (12, 14), (13, 14)}
   Here R3 consists of Ordered Pairs of A × B, whose first coordinate is less than the Second coordinate.
   Hence the formula for the relation R3 is ‘ Is Less than ’.

Set builder form of a relation:

We can represent the relations in set builder form also. In this method we describe the relation by stating the property that connects the first and second coordinate of every ordered pair of the relation

Example: If A = {2,3,4} then find A × A and express the following relations in Set Builder form ? E = { (2, 2), (3, 3), (4, 4)} and L = {(2, 3), (3, 4), (2, 4)} ?

Sol: Given A = {2, 3, 4}

        A × A = {2, 3, 4} × {2, 3, 4}

        A × A = {(2, 2), (2, 3), (2, 4) ,(3, 2), (3, 3) ,(3, 4), (4, 2), (4, 3), (4, 4)}

        Given relations are E = { (2, 2), (3, 3), (4, 4)}, L = {(2, 3), (3, 4), (2, 4)}

        Now, E = {(2, 2), (3, 3), (4, 4)}

        Here E consists of ordered pairs of A × A whose, first and second coordinates

        are equal.

        Set builder form for the relation E = {(2, 2), (3, 3), (4, 4)} is

                E = {(p, q)/(p, q) A × A, p = q}

        Now, L = {(2, 3), (3, 4), (2, 4)}

        Here L consists of ordered pairs of A × A, whose first coordinate is less

        than the second coordinate.

        Set Builder form for the Relation L = {(2, 3), (3, 4), (2, 4)} is

            L = {(p, q)/(p, q) A × A, p < q}

Inverse Relation:

If R is a relation from a set A into another set B, then by interchanging the first and second coordinates of ordered pairs of R we get a new relation from B into A. This relation is called the inverse relation. It is denoted by R–1.

That is for (x, y) R     (y, x) R–1.

The set builder form for the relation R is R–1 = {(y, x) / (x, y) R) }.

Example 1: If R = {(2, 3), (2, 4), (3, 4), (4, 3), (3, 2), (4, 2)} then find R–1 ?

Sol: Given relation R = {(2, 3), (2, 4), (3, 4), (4, 3), (3, 2), (4, 2)}

        We can observe that the domain of the relation R = {2, 3, 4}

        The range of the relation R = {2, 3, 4}

        R is a relation on the set {2, 3, 4} to {2, 3, 4}

        The inverse of the relation R–1 = {2, 3, 4} × {2, 3, 4}.

        R–1 = {(2, 3), (2, 4), (3, 4), (4, 3), (3, 2), (4, 2)}.

                    R = R–1.

Example 2: If A = {p, q, r}, B = {p, m, n} and R is a relation from A into B, then find R–1 ?

Sol: Given Sets are A = {p, q, r} , B = {p, m, n}

        R is a Relation from A B       R = {p, q, r} × {p, m, n}

            R = {(p, p), (p, m), (p, n), (q, p), (q, m), (q, n), (r, p) ,(r, m), (r, n)}.

        By the definition of the Inverse relation if we interchange the first and second

       coordinates of the relation R from A to B we will get the relation R–1 from B to A.

            R–1 = B × A = {p, m ,n} × { p, q , r}

            R–1 = {(p, p), (p, q), (p, r), (m, p), (m, q), (m, r), (n, p), (n, q), (n, r)}

Note:

The domain and range of R–1 are respectively range and domain of R.

For above example we can write that

            Domain of the relation R = A

            Range of the relation R = B

            Domain of the relation R–1 = B

            Range of the relation R–1 = A

Types of Relations

There are five types of relations namely:

1. Reflexive relation
2. Symmetric relation
3. Anti–symmetric relation
4. Transitive relation
5. Equivalance relation

Reflexive relation:

R is a relation in the set A (i.e. R A × A) and for every a A , (a, a) R, then R is said to be reflexive relation.

Examples:

1. Every real number is equal to itself. Implies ‘is equal to ’ is a reflexive relation in the set of real numbers.
   
2. The relation R = {(a, a), (b, b), (c, c)} in A = {a, b, c} is reflexive.
   
   Given Set is A = {a, b, c}
   
           A × A = {a, b, c} × {a, b, c}
   
           A × A = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}.
   
   Given relation is R = {(a, a), (b, b), (c, c)}
   
           For a, b, c A         (a, a), (b, b), (c, c) R.
   
           R is a Reflexive Relation.
3. l1, l2 are two straight lines in a plane. R is relation in the set of lines in the plane defined by l1 is parallel to l2 if (l1 , l2) R.
   
   Then R is reflexive as every line is parallel to itself.

Symmetric relation:

R is a relation in the set A (i.e., R A × A) and (a, b) R implies (b, a) R then R is said to be a symmetric relation.

Examples:

1. In the set of all real numbers " is equal to " relation is symmetric.
   
2. A is the set of lines in a plane. R is the relation in A defined by ‘is parallel ’, then R is symmetric. For if L1 , L2 are two lines in A and (L1 , L2) R then we have L1 || L2. But this implies that L2 || L1.
   
           (L2 , L1) R . Thus R is symmetric.

            So, R is a symmetric relation under ‘is parallel ’.

Anti –symmetric relation:

R is a relation in the set A (i.e., R A × A). If (a, b) R and (b, a) R implies

a = b, then R is said to be anti –symmetric relation.

Example:

1. In the set of all natural numbers in the relation R defined by " x divides y if and only if (x, y) R " is an anti –symmetric. If ‘x ’ divides ‘y ’ and ‘y ’ divides ‘x ’ then x = y.
2. In the set of all real numbers the relation " ≥ " is an anti – symmetric relation. For p ≥ q and q ≥ p imply p = q for p, q R.

Transitive relation:

R is a relation in the set A (i.e., R A × A). If (a, b) R and (b, c) R implies

(a, c) R, then the relation ‘R ’ is called transitive relation.

Examples:

1. In the set of real numbers the relation ‘is equal to ’ is a transitive relation. i.e., For a = b and b = c a = c.
2. If A = {1, 2}, B = {1, 2, 3} and C = {1, 2, 3, 4} then "is subset" is a transitive relation on these three sets.
   From above three sets we can observe that A B and B C     A C.
   
3. A is the set of all lines in a plane. R is the relation ‘is parallel to ’ in A.

            From above figure L1 || L2 and L2 || L3 then L1 || L3

              For (L1, L2) R , (L2, L3) R Implies (L1, L3) R.

            Hence the relation ‘is parallel to ’ is a transitive relation.

Equivalance relation:

R is a relation in the set A (i.e., R A × A), R is said to be ‘equivalance relation ’, if the relation is reflexive, symmetric and transitive.

Example: "=" is an equivalence relation i.e., it is reflexive, symmetric, transitive.

      Reflexive: Clearly, it is true that a = a for all values of a. So = is reflexive.

      Symmetric: If a = b, it is also true that b = a. So = is symmetric

      Transitive: If a = b and b = c, this says that a is the same as b which in turn

      is the same as c. So a is then the same as c, so a = c. Thus = is transitive.

Thus = is an equivalence relation. We denote an equivalence relation by symbol.

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