ICSE Class 7 Maths Chapter 21 Relations and Mapping

Chapter 21: Relations And Mapping

Ordered Pair

An ordered pair means a pair of two objects which occur in a particular order.

For example:

If two objects x and y are written as (x, y), they form one ordered pair and if written as (y, x), they form another ordered pair.

Since the order of writing the objects in both the pairs is different, therefore, they form two different ordered pairs.

If the ordered pairs are enclosed within curly braces (brackets), then we call it a set of ordered pairs.

For example:

Consider the following sets of ordered pairs:

(i) Persons and their ages: {(Meeta, 22), (Akshay, 20), (Sonu, 27)}

(ii) Numbers and their squares: {(4, 16), (5, 25), (6, 36), ......}

In an ordered pair (x, y), x is called the first component and y is called the second component.

1. Both the components of an ordered pair can be same - i.e., ordered pairs can be of the form (x, x), (y, y), (3, 3), etc.

2. Two ordered pairs (a, b) and (c, d) are equal, if a = c and b = d - e.g., (x, y) = (4, 2) if x = 4 and y = 2.

Example 1

Given: (a - 2, b + 1) = (1, 3), find a and b.

Solution:

(a - 2, b + 1) = (1, 3) implies a - 2 = 1 and b + 1 = 3

implies a = 1 + 2 and b = 3 - 1

implies a = 3 and b = 2 (Ans.)

Example 2

Given (4, y - 3) = (x + 2, 7), find x and y.

Solution:

(4, y - 3) = (x + 2, 7) implies 4 = x + 2 and y - 3 = 7

implies 4 - 2 = x and y = 7 + 3

implies 2 = x and y = 10

implies x = 2 and y = 10 (Ans.)

Teacher's Note

Ordered pairs appear in everyday life when we reference coordinates on maps or record test scores paired with student names. Understanding that (3, 5) is different from (5, 3) is crucial for navigation and data organization.

Relation

The word relation means an association of two objects (numbers, persons, etc.) based on some property connecting them.

For example:

(i) Peter is the son of John. This statement shows a relation between two persons. In this statement, the relation (R) means "is the son of".

(ii) 3 is a factor of 12. This statement shows a relation between two numbers. The relation (R) being "is a factor of".

Representation Of A Relation

1. Roster Form (as the set of ordered pairs):

Example 3

Given A = {1, 3, 4, 5, 6, 9, 10}, B = {0, 1, 2, 3} and a relation (R) from set A to set B means "is square of". Represent the relation 'R' in roster form.

Solution:

A relation from set A to set B means, the first component of each ordered pair is to be taken from set A and the second component from set B, such that the first component is square of the second component.

∴ R = {(1, 1), (4, 2), (9, 3)} (Ans.)

2. By Arrow Diagram:

In this method, the arrows are drawn from set A to set B to indicate pairing, which satisfy the given relation.

Example 4

Given ordered pairs: (6, 3), (6, 4), (5, 3), (5, 4), (4, 4), (3, 3), (3, 4), (3, 5) and (5, 5). Use these ordered pairs to find the following relations:

(i) R₁ = "is two more than"

(ii) R₂ = "is greater than"

(iii) R₃ = "is equal to".

Solution:

(i) R₁ = Set of all ordered pairs, whose first component is two more than the second component.

= {(6, 4), (5, 3)} (Ans.)

(ii) R₂ = Set of all ordered pairs, whose first component is greater than the second component.

= {(6, 3), (6, 4), (5, 3), (5, 4)} (Ans.)

(iii) R₃ = Set of all ordered pairs, whose first component is equal to the second component.

= {(4, 4), (3, 3), (5, 5)} (Ans.)

Example 5

Given ordered pairs: (2, 5), (3, 5), (2, 4), (3, 4), (2, 3), (3, 3), (2, 2), (3, 2), (4, 2), (4, 3), (4, 4) and (4, 5).

Use these ordered pairs to find the following relations:

(i) R₁ = "is less than"

(ii) R₂ = "is equal to"

(iii) R₃ = "is two less than"

Solution:

(i) R₁ = "is less than" = Set of all ordered pairs whose first component is less than the second component.

= {(2, 5), (3, 5), (2, 4), (3, 4), (2, 3), (4, 5)} (Ans.)

(ii) R₂ = "is equal to" = Set of all ordered pairs whose first component is equal to the second component.

= {(3, 3), (2, 2), (4, 4)} (Ans.)

(iii) R₃ = "is two less than" = Set of all ordered pairs whose first component is two less than the second component.

= {(3, 5), (2, 4)} (Ans.)

Example 6

Given A = {2, 4, 6}, B = {4, 18, 27} and a relation R from set A to set B means, "is a factor of". Find all ordered pairs which satisfy the given relation R. Represent the given relation by an arrow diagram.

Solution:

R = "is a factor of" = Set of all ordered pairs whose first component is from set A and second component is from set B in such a way that the first component of each ordered pair is a factor of the second component, i.e., the first component completely divides the second component.

= {(2, 4), (2, 18), (4, 4), (6, 18)} (Ans.)

Required arrow diagram is: [Arrow diagram showing: 2 points to 4 and 18; 4 points to 4; 6 points to 18]

Domain And Range Of A Relation

Domain: The set of first components of all ordered pairs of a relation is called its domain.

In Example 4, given above:

(i) the domain of relation R₁ = {6, 5},

(ii) the domain of relation R₂ = {6, 5},

and, (iii) the domain of relation R₃ = {4, 3, 5}.

Range: The set of second components of all ordered pairs of a relation is called its range.

In Example 4, given above:

(i) the range of relation R₁ = {4, 3},

(ii) the range of relation R₂ = {3, 4},

and, (iii) the range of relation R₃ = {4, 3, 5}.

Teacher's Note

Domain and range help us understand the scope of relationships - like how a teacher's grades (domain) map to student names (range), showing which students achieved which performance levels.

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