Class 8 – Relations

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Introduction: In our daily life, we come across many relations such as father-son, brother-sister, teacher-student and many more. In mathematics too, we come across relation such as A is a subset of B, Line l is parallel to line m. Number m is less than number n

In all these situations, we notice that a relation involves pairs of objects in certain order. This synopsis deals with the study of relations in mathematics.

 

Ordered pair:

In mathematics, an ordered pair is a combination of two objects.

Two elements a and b listed in a specific order, form an ordered pair, denoted by (a, b).

In an ordered pair (a, b), we call a as first component or first coordinate and b as second component or second coordinate. By changing the position of the components, the ordered pair is changed.

Thus, (a, b) ¹ (b, a)

 

Equality of ordered pairs:

Two ordered pairs (a, b) and (c, d) are said to be equal if and only if a = c and b = d.

i.e., (a, b) = (c, d)

Example: Given  Find a and b ?

Solution:

        

 

Cartesian product of two sets: For the given two sets A and B, the set containing all the ordered pairs where the first element is taken from A and second element is taken from B is called the Cartesian product of two sets A and B.

Cartesian product of two sets is denoted by  and read as A cross B.

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe set builder form

 

Example: Let

then, 

 

Example: Let A= {1,2,3}, B = {a, b}
A × B = {(1,a),(1,b)(2,a),(2,b),(3,a),(3,b)}

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

We notice from the above example that

i.e

further

= 3 × 2 =6

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i) A × B≠ B ×A unless A = B or A = Φ or B = Φ

ii) If n(A)= m; n(B) = n then n(A × B) = mn

iii) n(A ×B) = n(B × A)

iv) A × Φ = Φ × A = Φ

v) If A × B = Φ then either
a) A = Φ b)B = Φ c)A = Φ and B = Φ

Representation of Cartesian product:

Cartesian product of sets can be represented in the following ways.

1. Arrow diagram 2. Tree diagram 3.Graphical representation
4. Roster form 5. Set builder form

Representation of A×B in Roster form: In this form of representation, the elements of A × B (ordered pairs) are listed and each separated from the other by means of a comma.

Example: Let A= { p, q, r } B = { 2, 4 } then find A×B in Roster form

http://images.clipartpanda.com/bloc-clipart-note-md.pngIn the above example A×B can also be written in set builder form as

Representation of A × B in arrow diagram: If A= {x, y, z} and B = {1, 3, 5} then find A × B. In order to find A × B, represent the elements of A and B as shown in the diagram below:

Now draw the arrows from each element of A to each element of B, as shown in the above figure.

Now represent all the elements related by arrows in ordered pairs in a set, which is the required Cartesian product A × B.

A × B= {(x, 1), (x, 3), (x, 5), (y, 1), (y, 3), (y, 5), (z, 1),(z, 3), (z, 5)}

Representation of A×B using a tree diagram:

Example: If A = {2, 4, 6}, B = {1, 3, 5} thenfind A × B using tree diagram.
To represent A × B using tree diagram, write all the elements of A vertically and then for each element of A, write all the elements of B as shown and arrows as shown in the figure below.

∴ A ×B = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)}

Graphical Represention of A × B:

Example: Let A= {1, 3, 4} B = {2, 5, 6} then find A × B.

Considerthe elements of A along the X-axis and the elements of B along the

Y-axisand mark the points.

∴ A ×B = {(1,2), (1,5), (1,6), (3,2), (3,5), (3,6), (4,2), (4,5), (4,6)}

Proof: We have

Relation: Let A and B be two non-empty sets. Then a relation R from A to B is the set of all ordered pairs such that and a is associated with b according to a definite rule. If we say that is related to b and we write .

Clearly, every relation R from A to B is a subset of A × B. i.e., .

A“relation” is just a relationship between any two sets of information. Consider a relationship between all the people in a class and their heights. The pairing of names and heights is a relation. This pairing is done so that either if a person’s name is known we can give that person’s height, or else a height is given, the names of all the people who are that tall can be known. We also come across various relations in real life like ‘is mother of ’, is son of’,‘is teacher of ’. Let us now define the mathematical relation.

Definition: Let A and B be two non-empty sets and R is called a relation from the set A to B. (Any subset of is called a relation from A to B).

∴ A relation contains ordered pairs as elements. Hence “A relation is a set of ordered pairs”.

Example:

1.Let A= {1, 3, 4}, B = {1, 2}

Then,A × B = {(1,1), (1,2), (3,1), (3,2), (4,1), (4,2)}

Let R1= {(3, 1), (3, 2), (4, 1), (4, 2)}

We can observe that and also that forevery ordered pair (a, b) ∈ R1, a>b.

Hence R1 is the relation “is greater than” from A to B.

2. In the previous example, let Here also, and we also notice that for every ordered pair .

Hence, R2 is the relation “is less than or equal to” from A to B.

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i) If then the number of possible relations from A to B is 2mn. [Since every relation from A to B is a subset of A × B, the number of relations is equal to number of subsets of A × B].

ii) A relation R is said to be defined on a set A if .

iii) If n(A) = the number of relations defined on A, i.e.,

Example: Let A = {1, 2, ………10} then there are relations on A.

Domain and Range of a Relation: Let A  and B be two non-empty sets and R be a relation from A to B, we note that

i) The set of first co-ordinates of all ordered pairs in R is called the domain of R.

ii) The set of second co-ordinates of all ordered pairs in R is called the range of R.

Example: Let A = {1, 2, 4}, B = {a, b, c} and

R ={(1, a), (1, b), (2, a), (2, c), (4, c)} be a relation from A to B.

Then,domain of R = {1, 2, 4} and range of R = {a, b, c}

Representation of Relation:

i) Roster-method (or) List method: In this method we list all the ordered pairs that satisfy the rule or property given in the relation.
Example:
Let A = {1, 3, 5}, B = {3, 4, 6}
If R is a relation from A to B having property “is less than” then the roster form of R is
R = {(1, 3), (1, 4), (1, 6), (3, 4), (3, 6), (5, 6)}

ii) Set-builder method: In this method, a relation is described by using a representation and stating the property or properties, which the first and second co-ordinates of every ordered pair of the relation satisfy through the representation.
Example:
Let A = {1, 2, 4}
If R is a relation on A defined as “is greater than or equal to”, the set builder form of R is .

iii) Arrow diagram: In this method, a relation is described by drawing arrows between the elements which satisfy the property or properties given in the relation.
Example:
Let A = {2, 3, 4} B = {1, 3, 4}
Let R be a relation from A to B with property “is equal to”.
The arrow diagram of R is

http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

Let A and B be two non-empty sets. The Cartesian product of A and B, denoted by A · 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,b B}
1. A · B ≠ B · A, unless A = B
2. For any two sets A and B. n(A · B) = n(B · A)
3. If n(A) = p and n(B) = q, then n(A · B) = pq

If A = {1, 2, 3} and B = {3, 4, 5}, then find A · B.
Consider the elements of A on the X-axis and the elements of B on the Y-axis and mark the points