ICSE Class 8 Maths Algebra Chapter 23 Relations and Mappings

Relations And Mappings

Relations

There are 7 articles in a shop's godown which had been bought for Rs {90, 100, 110, 120, 130, 140, 150} in that order. Let us call this Set Cp.

There are 7 articles on display in the showroom, the selling prices of which are Rs {95, 105, 115, 125, 135, 145, 155} respectively. Let us call this Set Sp.

If it is known that all articles in the shop are sold at a profit of Rs 15, we can easily pair off 6 articles from Cp and Sp that match as:

Set R = {(90, 105), (100, 115), (110, 125), (120, 135), (130, 145), (140, 155)}. Notice that one article in the godown that cost Rs 150 is not displayed and one article on display that is selling for Rs 95 is not in the godown. The set of ordered pairs R is such that there exists, between each first component and second component, a relation from Cp to Sp such that S = C + 15.

Similarly, in the set of ordered pairs R1 = {(105, 90), (115, 100), (125, 110), (135, 120), (145, 130), (155, 140)} there exists a relation from Sp to Cp, such that C = S - 15.

The set of first components in a relation is known as its domain.

Domain of R = {90, 100, 110, 120, 130, 140},
Domain of R1 = {105, 115, 125, 135, 145, 155}.

The set of second components in a relation is known as its range.

Range of R = {105, 115, 125, 135, 145, 155}
Range of R1 = {90, 100, 110, 120, 130, 140}.
Although {95, 150} is an ordered pair, it does not belong to either R or R1.

Example 1: Given:

A = {-25, -15, -5, 0, 5, 15, 25}
B = {-30, -20, -10, 0, 10, 20, 30}

(i) Write relation R1 from A to B describing 'is more than'.

We look at each term in A and make ordered pairs with a term in B that is less in value.

R1 = {(-25, -30), (-15, -30), (-15, -20), (-5, -30), (-5, -20), (-5, -10), (0, -30), (0, -20), (0, -10), (5, -30), (5, -20), (5, -10), (5, 0), (15, -30), (15, -20), (15, -10), (15, 0), (15, 10), (25, -30), (25, -20), (25, -10), (25, 0), (25, 10), (25, 20)}

(ii) Write relation R2 from A to B describing 'is 5 more than.'

R2 = {(-25, -30), (-15, -20), (-5, -10), (5, 0), (15, 10), (25, 20)}

(iii) Write relation R3 from B to A describing 'is half of its sum with 5' and write its domain and range.

The first components of R3 will come from Set B as the relation from B to A is \(\frac{a + 5}{2}\).

(R3 = {(-10, -25), (0, -5), (10, 15)})
where domain = {-10, 0, 10}
and range = {-25, -5, 15}

The relation between different sets is graphically represented by arrow diagrams. The three relations in the above example are described by the following arrow diagrams.

Mapping

If every element in Set A is related to one and only one element in Set B, the relation from A to B is called a mapping.

Remember

For a relation from A to B to be a mapping,
- the domain of the relation must be equal to Set A.
- no two ordered pairs in the relations should have the same first component.

Thus, the relation between the cost price and selling price of the articles in the shop at the beginning of this chapter is not a mapping as the article costing Rs 150 was not related to any article displayed in the showroom.

Example 2: Given A = {2, 3, 4, 5} and B = {2, 3, 4, 5, 6, 7, 8, 9, 10}, which of the following relations from A to B are mappings?

(i) R1 = {(3, 3), (4, 4), (5, 5)}
The element 2 is not present in the domain of R1. Thus R1 is not a mapping.

(ii) R2 = {(2, 4), (3, 6), (4, 8), (5, 10)}
The domain of R2 is equal to Set A and no two ordered pairs have the same first component. Thus R2 is a mapping.

(iii) R3 = {(2, 2), (2, 4), (2, 6), (2, 8), (2, 10), (3, 3), (3, 6), (3, 9), (4, 4), (4, 8), (5, 5), (5, 10)}
The domain of R3 is equal to set A but the elements in the domain do not have unique second components. Also more than two ordered pairs have the same first components. Thus R3 is not a mapping.

(iv) R4 = {(2, 5) (3, 5), (4, 9), (5, 9)}
The domain of R4 is equal to Set A and although the second components are repeated, no two ordered pairs have the same first component. Thus R4 is a mapping.

Teacher's Note

Relations and mappings help us understand how different quantities connect in real life, such as how students' names map to their unique student ID numbers, where each student has exactly one ID.

Exercise 23.1

1. Figure out a suitable relation between sets A and B in each of the following and draw arrows from A to B.

(i) Set A contains: Dog, Cat, Pig, Lion, Crow, Horse. Set B contains: Caw, Roar, Neigh, Mew, Fly, Bark.

(ii) Set A contains: Stable, Coop, Sty, Barn, Hut, Nest. Set B contains: Cow, Man, Horse, Chicken, Bird, Pig.

(iii) Set A contains: 5, 3, 7, 9, 6. Set B contains: 6, 14, 10, 12, 18.

(iv) Set A contains: 2, 3. Set B contains: 4, 9, 16, 81, 27, 8.

2. Given A = {6, 8, 10, 12, 14} and B = {5, 7, 9, 11, 13}, does R = {(5, 8), (7, 10), (9, 12), (11, 14)} describe a relation from A to B or B to A? Describe the relation.

3. Use the following ordered pairs to write the relation as indicated: (6, 18), (10, 2), (20, 4), (35, 7), (36, 9), (9, 81), (7, 21), (3, 9), (7, 49)

(i) R1 describing 'is a multiple of'

(ii) R2 describing 'is a factor of'

(iii) R3 describing 'is one-third of'

(iv) R4 describing 'is the square root of'

(v) R5 describing 'is five times'

4. Given A = {11, 9, 0, 25, 5} and B = {5, 2, -3, \(-\frac{1}{2}\), \(\frac{4}{5}\)}, write the following relations in Roster form and represent them by arrow diagrams.

(i) R1 describing a = 5b

(ii) R2 describing a = 2b + 6

(iii) R3 describing a = 2b + 1

(iv) R4 describing a = 3b - 1

(v) R5 describing a = b2

Teacher's Note

Understanding different types of relations through ordered pairs helps students recognize patterns in everyday situations, such as recognizing that every person has one unique fingerprint, establishing a clear mapping between individuals and their biometric data.

Exercise 23.2

1. Which of the following arrow diagrams represent mappings?

(i) Set A contains: 5, 10, 15, 30. Set B contains: 50, 450. All elements in A map to 50 or 450.

(ii) Set A contains: 25, 64. Set B contains: 5, -5, 8, -8.

(iii) Set A contains: 5, 15, 25, 35, 45. Set B contains: 17, 27, 37, 47, 57.

(iv) Set A contains: 9, 10, 11, 12, 13. Set B contains: 10, 11, 12, 13, 14.

2. Given Set A = {a, b, c, d, e} and Set B = {22, 23, 24, 25, 26},

(i) write a relation R1 from A to B that is not a mapping.

(ii) write a relation R2 from B to A that is not a mapping.

(iii) write a relation R3 from A to B that is a mapping.

(iv) write a relation R4 from B to A that is a mapping.

3. The distance covered by a bus driver over a week is given in Table 23.1. Is the relation between his average speed and the distance covered each day a mapping? Explain why. Write the relation in Roster form and draw an arrow diagram.

4. A man invested different amounts of money in various deposit schemes. Over a period of 1 year, the interest earned on them was as follows:

Is the relation between the rates of interest on the various deposit schemes and the principal amounts invested in them a mapping? Explain why. Write the relation in Roster form and draw an arrow diagram.

Teacher's Note

Mappings in practical contexts like investment returns help students see why banks need consistent interest rate calculations - each principal amount should map to exactly one interest rate to ensure fairness and clarity in financial transactions.

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