# Example 43 - Chapter 1 Class 12 Relation and Functions (Term 1)

Last updated at Jan. 28, 2020 by Teachoo

Examples

Example 1

Example 2

Example 3

Example 4 Important

Example 5

Example 6 Important

Example 7

Example 8 Important

Example 9

Example 10

Example 11 Important

Example 12 Important

Example 13 Important

Example 14 Important

Example 15 Deleted for CBSE Board 2022 Exams

Example 16 Deleted for CBSE Board 2022 Exams

Example 17 Deleted for CBSE Board 2022 Exams

Example 18 Important Deleted for CBSE Board 2022 Exams

Example 19 Important Deleted for CBSE Board 2022 Exams

Example 20 Deleted for CBSE Board 2022 Exams

Example 21 Deleted for CBSE Board 2022 Exams

Example 22 Deleted for CBSE Board 2022 Exams

Example 23 Important Deleted for CBSE Board 2022 Exams

Example 24 Deleted for CBSE Board 2022 Exams

Example 25 Important Deleted for CBSE Board 2022 Exams

Example 26 Deleted for CBSE Board 2022 Exams

Example 27 Important Deleted for CBSE Board 2022 Exams

Example 28 (a) Deleted for CBSE Board 2022 Exams

Example 28 (b) Deleted for CBSE Board 2022 Exams

Example 28 (c) Deleted for CBSE Board 2022 Exams

Example 29 Deleted for CBSE Board 2022 Exams

Example 30 Deleted for CBSE Board 2022 Exams

Example 31 Important Deleted for CBSE Board 2022 Exams

Example 32 Deleted for CBSE Board 2022 Exams

Example 33 Deleted for CBSE Board 2022 Exams

Example 34 Deleted for CBSE Board 2022 Exams

Example 35 Deleted for CBSE Board 2022 Exams

Example 36 Deleted for CBSE Board 2022 Exams

Example 37 Important Deleted for CBSE Board 2022 Exams

Example 38 Deleted for CBSE Board 2022 Exams

Example 39 Deleted for CBSE Board 2022 Exams

Example 40 Deleted for CBSE Board 2022 Exams

Example 41

Example 42 Important

Example 43 Important You are here

Example 44

Example 45 (a) Deleted for CBSE Board 2022 Exams

Example 45 (b) Deleted for CBSE Board 2022 Exams

Example 46 Important Deleted for CBSE Board 2022 Exams

Example 47 Important Deleted for CBSE Board 2022 Exams

Example 48 Important Deleted for CBSE Board 2022 Exams

Example 49 Deleted for CBSE Board 2022 Exams

Example 50

Example 51 Important

Chapter 1 Class 12 Relation and Functions (Term 1)

Serial order wise

Last updated at Jan. 28, 2020 by Teachoo

Example 43 (Method 1) Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation in X given by R1 = {(x, y) : x – y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}. Show that R1 = R2. R1 = {(x, y) : x – y is divisible by 3} R2 = { (x, y): {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} } We will prove R1 = R2 by proving R1 ⊂ R2 and R2 ⊂ R1 i.e, all elements of R1 are in the set R2 and all elements of R2 are in the set R1 Proving R1 ⊂ R2 Let (x, y) ∈ R1 ⇒ x – y is a divisible 3 ⇒ {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ (x, y) ∈ R2. Hence, R1⊂ R2. Rough In {1, 4, 7} x – y = 1 – 4 = – 3 = 4 – 7 = –3 = 7 – 1 = 6 So, x – y is divisible by 3 Similarly, in {2, 5, 8} & {3, 6, 9} x – y is divisible by 3 Proving R2 ⊂ R1 Let (x, y) ∈ R2 ⇒ {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ x – y is divisible 3 ⇒ (x, y) ∈ R1. Hence, R2 ⊂ R1. Hence, R1 ⊂ R2 & R2 ⊂ R1. ∴ R1 = R2 Hence shown Rough In {1, 4, 7} x – y = 1 – 4 = 3 = 4 – 7 = –3 = 7 – 1 = 6 So, x – y is divisible by 3 Similarly, in {2, 5, 8} & {3, 6, 9} x – y is divisible by 3 Example 43 (Method 2) Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation in X given by R1 = {(x, y) : x – y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): {x, y} ⊂ {1, 4, 7}} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}. Show that R1 = R2. X = {1, 2, 3, 4, 5, 6, 7, 8, 9} R1 = {(x, y) : x – y is divisible by 3} Finding R1 R1 = { (1, 4), (1, 7) , (2, 5) , (2, 8), (3, 6) , (3, 9) , (4, 1), (4, 7), (5, 2), (5, 8) , (6, 3) , (6, 9), (7, 1) , (7, 4) , (8, 2), (8, 5), (9, 3), (9, 6) } Now, finding R2 R2 = { (x, y): {x, y} ⊂ {1, 4, 7} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} } R2 = { (1, 4), (4, 1) , (1, 7) ,(7, 1), (4, 7) , (7, 4) , (2, 5), (5, 2) , (2, 8) ,(8, 2), (5, 8) , (8, 5) , (3, 6), (6, 3), (3, 9) , (9, 3), (6, 9) , (9, 6) } ∴ R1 = R2 Hence proved