Important Key Points

Example 1.4
Let A = {3,4,7,8} and B = {1,7,10}. Which of the following sets are relations from A to B?
(i) $R_{1}$ ={(3,7), (4,7), (7,10), (8,1)}
(ii) $R_{2}$= {(3,1), (4,12)}
(iii) $R_{3}$= {(3,7), (4,10), (7,7), (7,8), (8,11), (8,7), (8,10)}

Solution :
To determine whether the given set is a relation, first we have to find the cartesian product of two sets.
A$\times$B = {(3,1), (3,7), (3,10), (4,1), (4,7), (4,10), (7,1), (7,7), (7,10), (8,1), (8,7), (8,10)}
(i) To say $R_{1}$ is the relation , every element of $R_{1}$ should belongs to A$\times$B. We note that, R1$\subseteq$ A×B . Thus, $R_{1}$ is a relation from A to B.
(ii)To say $R_{2}$ is the relation , every element of $R_{2}$ should belongs to A$\times$B. But (4,12) $\notin$ A×B . So, $R_{2}$ is not a relation from A to B.
(iii) To say $R_{3}$ is the relation , every element of $R_{3}$ should belongs to A$\times$B.but (7, 8), (8,11) $\notin$ A×B . So, $R_{3}$ is not a relation from A to B.

Example 1.5
The arrow diagram shows (Fig.1.10) a relationship between the sets P and Q. Write the relation in
(i) Set builder form
(ii) Roster form
(iii) What is the domain and range of R.

Solution :
(i) Set builder form of R = {(x,y) | y = x −2, x $\in$ P,y $\in$Q}
(ii) Roster form R = {(5, 3),(6, 4),(7,5)}
(iii) Domain of R = {5,6,7} and range of R = {3,4,5} (Note : Domain of R differs from Domain of P)

Exercise 1.2

1. Let A = {1, 2, 3, 7} and B = {3, 0, -1, 7}, which of the following are relation from A to B?
(i) R1 = {(2,1), (7,1)}
(ii) R2 = {(-1,1)}
(iii) R3 = {(2,-1), (7, 7), (1,3)}
(iv) R4 = {(7, -1), (0, 3), (3, 3), (0, 7)}

Solution :
To determine whether the given set is a relation, first we have to find the cartesian product of two sets.
Given A = {1,2,3,7} B = {3,0,-1, 7}
A × B = {(1,3) (1,0) (1,-1) (1,7) (2,3) (2, 0) (2, -1) (2, 7) (3, 3) (3,0) (3,-1) (3, 7) (7, 3) (7, 0) (7,-1) (7, 7)}
(i) To say $R_{1}$ is the relation , every element of $R_{1}$ should belongs to A$\times$B. But (2,1), (7,1) $\notin$ A$\times$B. So $R_{1}$ is not a relation from A to B.
(ii)To say $R_{2}$ is the relation , every element of $R_{2}$ should belongs to A$\times$B. But (-1,1)$\notin$ A$\times$B, Thus $R_{2}$ is not a relation from A to B.
(iii)To say $R_{3}$ is the relation , every element of $R_{3}$ should belongs to A$\times$B. Here every element of $R_{3}$ belongs to A$\times$B. i.e $R_{3}$ $\subseteq$ A$\times$B.Thus $R_{3}$ is a relation from A to B.
(iv)To say $R_{4}$ is the relation , every element of $R_{4}$ should belongs to A$\times$B. But (0,3), (0,7)$\notin$ A$\times$B.Thus $R_{2}$ is not a relation from A to B.

2. Let A = {1, 2, 3, 4,…,45} and R be the relation defined as “is square of ” on A. Write R as a subset of A × A. Also, find the domain and range of R.

Solution :
A = {1, 2, 3, 4, . . . 45},
A × A = {(1, 1), (2, 2),(3,3),...,(45, 45)}
R be the relation defined as “is square of ” on A. i.e 1 is related to $1^2=1$.
2 is related to $2^2=4$.
$\Rightarrow$ R = {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25), (6, 36)}
i.e R ⊂ (A × A)
Domain of R = {1, 2, 3, 4, 5, 6}
Range of R = {1, 4, 9, 16, 25, 36}

3. A Relation R is given by the set {(x, y)/y = x + 3, x ∈ {0, 1, 2, 3, 4, 5}}. Determine its domain and range.

Solution :
x = {0, 1, 2, 3, 4, 5}
y = x + 3
for x = 0 ⇒ y = 0 + 3 = 3
for x = 1 ⇒ y = 1 + 3 = 4
for x = 2 ⇒ y = 2 + 3 = 5
for x = 3 ⇒ y = 3 + 3 = 6
for x = 4 ⇒ y = 4 + 3 = 7
for x = 5 ⇒ y = 5 + 3 = 8
R = {(0, 3) (1,4) (2, 5) (3, 6) (4, 7) (5, 8)}
Domain = {0, 1, 2, 3, 4, 5}
Range = {3, 4, 5, 6, 7, 8}

4. Represent each of the given relations by
(a) an arrow diagram
(b) a graph and
(c) a set in roster form, wherever possible.
(i) {(x,y) | x = 2y,x ∈ {2, 3, 4, 5}, y ∈ {1, 2, 3, 4}
(ii) {(x, y) | y = x + 3, x, y are natural numbers < 10}

Solution :
(i) x = {2, 3, 4, 5} y = {1, 2, 3, 4}
x = 2y
for y = 1 ⇒ x = 2 × 1 = 2
for y = 2 ⇒ x = 2 × 2 = 4
for y = 3 ⇒ r = 2 × 3 = 6
for y = 4 ⇒ x = 2 × 4 = 8
(a) Arrow diagram
(b) Graph
(c) Roster form R = {(2, 1) (4, 3)}
(ii) x = {1, 2, 3, 4, 5, 6, 7, 8, 9}, y = {1,2, 3, 4, 5, 6, 7, 8,9}
y = x + 3
when x = 1 ⇒ y = 1 + 3 = 4
when x = 2 ⇒ y = 2 + 3 = 5
when x = 3 ⇒ y = 3 + 3 = 6
when x = 4 ⇒ y = 4 + 3 = 7
when x = 5 ⇒ y = 5 + 3 = 8
when x = 6 ⇒ y = 6 + 3 = 9
when x = 7 ⇒ y = 7 + 3 = 10
when x = 8 ⇒ y = 8 + 3 = 11
when x = 9 ⇒ y = 9 + 3 = 12
R = {(1,4) (2, 5) (3,6) (4, 7) (5, 8) (6, 9)}
(a) Arrow diagram
(b) Graph
(c) Roster form R = {(2, 1) (4, 3)}

5. A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (E). The company provide ?10,000, ?25,000, ?50,000 and ?1,00,000 as salaries to the people who work in the categories A, C, M and E respectively. If $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ and $A_{5}$ were Assistants; $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$ were Clerks; $M_{1}$, $M_{2}$, $M_{3}$ were managers and $E_{1}$, $E_{2}$ were Executive officers and if the relation R is defined by xRy, where x is the salary given to person y, express the relation R through an ordered pair and an arrow diagram.

Solution :
Assistants ? $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$
Clerks ? $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$
Managers ? $M_{1}$, $M_{2}$, $M_{3}$
Executive officers ? $E_{1}$, $E_{2}$
R = {10000, A1), (10000, A2), (10000, A3), (10000, A4), (10000, A5), (25000, C1), (25000, C2), (25000, C3), (25000, C4), (50000, M1), (50000, M2), (50000, M3), (100000, E1), (100000, E2)}
(a) Arrow diagram