ICSE Class 8 Maths Algebra Chapter 12 Relations and Mappings

Relations And Mappings

Relations

Arrow Diagrams

Let us consider the statement, "the tiger is the national animal of India and Bangladesh, the lion that of Belgium, and the Zebra that of Botswana". We can draw the following arrow diagram to represent it.

The animals (the elements of the first set) and the countries (the elements of the second set) are linked by a relation "is the national animal of". The arrows are drawn between the ordered pairs, or matching pairs, of elements that satisfy the relations.

A relation is a link between the elements of two sets based on some common property. In your previous class, you learnt two ways of representing relations - by making arrow diagrams and by expressing in the roster form.

Roster Form

Let us call the set of animals A and the set of countries B. To express the relation between the two sets in the roster form, we write the matching, or ordered pairs (Tiger, India), (Tiger, Bangladesh), (Lion, Belgium), (Zebra, Botswana). Then the pair (Tiger, India) means "the tiger is the national animal of India". Thus, we get

A = {Tiger, Lion, Zebra}, B = {India, Bangladesh, Belgium, Botswana} and

R = {(Tiger, India), (Tiger, Bangladesh), (Lion, Belgium), (Zebra, Botswana)},

where R is the relation from the set A to the set B.

Notice that while writing the ordered pairs we have written the elements of the set A first. This is because the relation is from the set A to the set B.

To express a relation from set A to set B in the roster form, write the elements of set A first. To express a relation from set B to set A, write the elements of set B first.

We could also write Tiger R India, Tiger R Bangladesh, Lion R Belgium, Zebra R Botswana. Here, R stands for "is the national animal of".

Thus R = {(a, b) : a ∈ A, b ∈ B and a is the national animal of b}.

Express The Following Relations In The Roster Form

Example 1 - Part (i)

The matching pairs are (1, 1), (4, 2), (9, 3), (16, 4), (25, 5).

So, the relation R = {(1, 1), (4, 2), (9, 3), (16, 4), (25, 5)} is from the set A = {1, 4, 9, 16, 25} to the set B = {1, 2, 4, 5, 3}. Also, in each pair the first element "is the square of" the second element.

We write: 1 R 1, 4 R 2, 9 R 3, 16 R 4, 25 R 5

and R = {(a, b) : a ∈ A, b ∈ B and a = b²}.

Example 1 - Part (ii)

The matching pairs are (Rupee, India), (Dollar, USA), (Yen, Japan) and (Lira, Italy).

So, the relation is R = {(Rupee, India), (Dollar, USA), (Yen, Japan), (Lira, Italy)} from the set A = {Rupee, Dollar, Yen, Lira} to the set B = {USA, India, Italy, Japan}, and it can be described as "is the currency of".

We can also write Rupee R India, Dollar R USA, Yen R Japan, Lira R Italy and R = {(a, b) : a ∈ A, b ∈ B and a is the currency of b}.

Example 2

Let A = {1, 2, 3, 4}, B = {1, 8, 27, 64} and R be the relation "is the cube root of" from the set A to set B. Represent R in the form of a arrow diagram and in the roster form.

Solution

Since 1 = ∛1, 2 = ∛8, 3 = ∛27 and 4 = ∛64 we write

1 R 1, 2 R 8, 3 R 27, 4 R 64.

So, the ordered pairs are (1, 1), (2, 8), (3, 27) and (4, 64). The arrow diagram is shown along side.

In the roster form, R can be written as follows.

R = {(1, 1), (2, 8), (3, 27), (4, 64)}

i.e., R = {(a, b) : a ∈ A, b ∈ B and a = ∛b}.

Example 3

Let A = {2, 5, 7, 11}, B = {1, 6, 15, 49} and R be the relation "is a factor of" from set A to set B. Represent R with the help of an arrow diagram and in the roster form.

Solution

2 is a factor of 6, 5 is a factor of 15, 7 is a factor of 49, i.e., 2 R 6, 5 R 15, 7 R 49.

So, the ordered pairs are (2, 6), (5, 15) and (7, 49). The arrow diagram is shown alongside.

In roster form, R can be written as follows.

R = {(2, 6), (5, 15), (7, 49)}

i.e., R = {(a, b) : a ∈ A, b ∈ B and a is a factor of b}.

Domain And Range

The set of all the first elements (or components) of the ordered pairs of a relation is called the domain of the relation. The set of all the second elements (or components) of the ordered pairs of a relation is called the range of the relation.

In the preceding two examples the domains are {1, 2, 3, 4} and {2, 5, 7}, while the ranges are {1, 8, 27, 64} and {6, 15, 49} respectively. Notice that in the first case, domain = set A and range = set B. But in the second case, domain ≠ set A and range ≠ set B. In fact, domain ⊂ set A and range ⊂ set B.

Example 4

Let A = {0, 1, 2, 3, 4, 5}, B = {1, 2, 3, 5, 7, 9} and R = {(a, b) : a ∈ A, b ∈ B and a + b = 8}.

Find R And Represent It By An Arrow Diagram

Clearly, 1 + 7 = 8, 3 + 5 = 8, 5 + 3 = 8.

So, (1, 7) ∈ R, (3, 5) ∈ R and (5, 3) ∈ R.

(i) R = {(1, 7), (3, 5) and (5, 3)}.

The arrow diagram for R is shown alongside.

Find The Domain And Range Of R

(ii) The domain of R = {1, 3, 5}.

The range of R = {3, 5, 7}.

Remember These

1. A set R of matching pairs or ordered pairs (x, y) where x ∈ A, y ∈ B, is a relation from the set A to the set B.

Thus, R = {(x, y) : x ∈ A, y ∈ B}.

2. To represent an ordered pair (x, y) in an arrow diagram, draw an arrow from x to y.

4. To form an ordered pair from a set A to set B on the basis of an arrow diagram, write the element from set A first, followed by the element of the set B to which the arrow points.

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