Class 9 – Relations

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Relations
  • Introduction to relations
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Relations
  • Types of relations
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Introduction: Study the following sentences:

1. Dashratha was father of Rama.

2. New Delhi is the capital of India. .

3. 52 is greater than 43.

4. The line AB is parallel to the line CD

5. {a, b} is a subset of {a, b, c}.

All these sentences express relationship between two objects. In mathematics, symbols replace phrases. Thus, the phrases is ‘is larger than’, ‘is parallel to’, ‘is subset of’, are respectively replaced by the symbols. ‘ > ’, ‘ ’, ‘ ’.

Ordered pair: A pair of mathematical objects a, b usually written in parentheses like (a, b) is called an ordered pair. In ordered pair (a, b) a and b are called the first and second coordinates or the left and right projections of the ordered pair.

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1) The order in which the objects appear in the pair is significant i.e., (a, b) ≠ (b, a)
unless a = b.
2) (a, b) = (c, d) _ a = c and b = d

Example: Find a and b if (a, 4) = (3, b)

(a, 4) = (3, b)

_ a = 3 and b = 4.

Cartesian product of two sets: Let A and B be two non-empty sets. The Cartesian product of A and B denoted by A _ B is defined as the set of all ordered pairs (a, b) such that a _ A and b _ B.

http://images.clipartpanda.com/bloc-clipart-note-md.png A _ B = {(a, b)/a _ A and b _ B}

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

we notice from the above example that

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1) unless A = B or

2) If; then

3)
4)

5) If then either
i) ii)
iii) and

A “relation”is just a relationship between any two sets of information. Consider arelationship 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 student of ’. Let us now define the mathematical relation.

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

_Arelation contains ordered pairs as elements. Hence “A relation is a set ofordered 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

We can observe that and also that for every ordered pair (a, b) , a > b.

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

http://images.clipartpanda.com/bloc-clipart-note-md.png 1. If and , then the number of possible relations from A to B is [since every relation from A to B is a subset of , the number of relations is equal to number of subsets of ].
2. A relation R is said to be defined as a set A if .
3. If the number ofrelations defined on A, i.e., A _ A is

Example: Let A = { 1, 2, ………10 }

There are relations onA.

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

(a) The set of first co-ordinates ofall ordered pairs in R is called the domain of R.

(b) The set of second co-ordinates ofall ordered pairs in R is called the range of R.

Example: Let A = {1, 2, 4}, B = {1, 2, 3} and

R = {(1, 1), (1, 2), (2, 1), (2, 3),(4, 3)} be a relation from A to B.

Then, domain of R = {1, 2, 4} and rangeof R = {1, 2, 3}

Representation ofRelation:

i) Roster-method (or) Listmethod: In this method we list all the ordered pairs that satisfy the rule or property given in the relation.
Example: LetA = {1, 3, 5}, B = {3, 4, 6}
If R is a relation from A to B having property “is less than” then the rosterform 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 thefirst and second co-ordinates of every ordered pair of the relation satisfy,through the algebraic representation.
Example: LetA = {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