Relations

NCERT Class 11 Maths • Exercise 2.2

1. Relation on Set A

Q1: $A = \{1, \dots, 14\}$. $R = \{(x, y) : 3x – y = 0\}$. Find D, R, C.

Relation: $3x – y = 0 \Rightarrow y = 3x$.
Since $x, y \in A$, we substitute values of $x$:
If $x=1, y=3$
If $x=2, y=6$
If $x=3, y=9$
If $x=4, y=12$
If $x=5, y=15$ ($15 \notin A$, Stop).

Roster Form: $R = \{(1, 3), (2, 6), (3, 9), (4, 12)\}$

Domain: $\{1, 2, 3, 4\}$
Range: $\{3, 6, 9, 12\}$
Codomain: $\{1, 2, \dots, 14\}$ (Set A)

2. Relation on Natural Numbers

Q2: $R = \{(x, y) : y = x + 5, x < 4, x \in N\}$. Write R, Domain, Range.

Logic: $x$ is a natural number less than 4.
Possible values for $x = \{1, 2, 3\}$.
$y = x + 5$.

When $x=1, y=6 \rightarrow (1,6)$
When $x=2, y=7 \rightarrow (2,7)$
When $x=3, y=8 \rightarrow (3,8)$

Roster Form: $R = \{(1, 6), (2, 7), (3, 8)\}$

Domain: $\{1, 2, 3\}$ Range: $\{6, 7, 8\}$

3. Odd Difference Relation

Q3: $A=\{1,2,3,5\}, B=\{4,6,9\}$. $R=\{(x,y): |x-y| \text{ is odd}\}$.

Tip: The difference between two numbers is odd if one is Even and the other is Odd.
Since $A$ has $\{1, 3, 5\}$ (Odds) and $\{2\}$ (Even).
$B$ has $\{9\}$ (Odd) and $\{4, 6\}$ (Evens).

Pairs must be (Odd, Even) or (Even, Odd).

Roster Form:
$R = \{ (1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6) \}$

4. Set-Builder & Roster from Figure

Q4: Describe the relationship between P and Q (Fig 2.7).

Observation from Figure (Standard NCERT):
$P = \{5, 6, 7\}, Q = \{3, 4, 5\}$
Pairs: $(5,3), (6,4), (7,5)$.
Pattern: $y = x – 2$.

(i) Set-Builder Form:
$R = \{(x, y) : y = x – 2, x \in P\}$

(ii) Roster Form:
$R = \{(5, 3), (6, 4), (7, 5)\}$

Domain: $\{5, 6, 7\}$
Range: $\{3, 4, 5\}$

5. Divisibility Relation

Q5: $A=\{1, 2, 3, 4, 6\}$. $R=\{(a,b) : b \text{ is exactly divisible by } a\}$.

We look for pairs $(a, b)$ where $a$ divides $b$.

(i) Roster Form:
$R = \{ \\ (1,1), (1,2), (1,3), (1,4), (1,6), \\ (2,2), (2,4), (2,6), \\ (3,3), (3,6), \\ (4,4), \\ (6,6) \\ \}$

(ii) Domain: $\{1, 2, 3, 4, 6\}$ (Set A)

(iii) Range: $\{1, 2, 3, 4, 6\}$ (Set A)

6 & 7. Logic Mapping

Q6: $R = \{(x, x+5) : x \in \{0, 1, 2, 3, 4, 5\}\}$. Find D & R.

Pairs: $(0,5), (1,6), (2,7), (3,8), (4,9), (5,10)$

Domain: $\{0, 1, 2, 3, 4, 5\}$ Range: $\{5, 6, 7, 8, 9, 10\}$

Q7: $R = \{(x, x^3) : x \text{ is prime } < 10\}$. Write in Roster.

Primes less than 10: $2, 3, 5, 7$.
$y = x^3$.

$2^3 = 8, 3^3 = 27, 5^3 = 125, 7^3 = 343$.

Answer:
$R = \{(2, 8), (3, 27), (5, 125), (7, 343)\}$

8. Calculating Relations

Q8: Let $A=\{x, y, z\}, B=\{1, 2\}$. Find number of relations from A to B.

Formula: If $n(A)=p$ and $n(B)=q$, total relations = $2^{pq}$.

$n(A) = 3$

$n(B) = 2$

$n(A \times B) = 3 \times 2 = 6$

Total Relations: $2^6 = 64$.

9. Relation on Integers (Z)

Q9: $R = \{(a, b) : a, b \in Z, a – b \text{ is an integer}\}$. Find D & R.

Logic: The difference of any two integers is always an integer.
$a – b \in Z$ for all $a, b \in Z$.

This means every integer is related to every other integer.

Domain: $Z$ (All Integers) Range: $Z$ (All Integers)