




Subscribe to our Youtube Channel - https://you.tube/teachoo
Last updated at Jan. 28, 2020 by Teachoo
Transcript
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
Examples
Example 2
Example 3
Example 4 Important
Example 5
Example 6 Important
Example 7
Example 8
Example 9
Example 10
Example 11 Important
Example 12 Important
Example 13 Important
Example 14 Important
Example 15 Not in Syllabus - CBSE Exams 2021
Example 16 Not in Syllabus - CBSE Exams 2021
Example 17 Not in Syllabus - CBSE Exams 2021
Example 18 Important Not in Syllabus - CBSE Exams 2021
Example 19 Important Not in Syllabus - CBSE Exams 2021
Example 20 Not in Syllabus - CBSE Exams 2021
Example 21 Not in Syllabus - CBSE Exams 2021
Example 22 Not in Syllabus - CBSE Exams 2021
Example 23 Important Not in Syllabus - CBSE Exams 2021
Example 24 Not in Syllabus - CBSE Exams 2021
Example 25 Important Not in Syllabus - CBSE Exams 2021
Example 26 Not in Syllabus - CBSE Exams 2021
Example 27 Important Not in Syllabus - CBSE Exams 2021
Example 28 Not in Syllabus - CBSE Exams 2021
Example 29 Not in Syllabus - CBSE Exams 2021
Example 30 Not in Syllabus - CBSE Exams 2021
Example 31 Not in Syllabus - CBSE Exams 2021
Example 32 Not in Syllabus - CBSE Exams 2021
Example 33 Not in Syllabus - CBSE Exams 2021
Example 34 Not in Syllabus - CBSE Exams 2021
Example 35 Not in Syllabus - CBSE Exams 2021
Example 36 Not in Syllabus - CBSE Exams 2021
Example 37 Not in Syllabus - CBSE Exams 2021
Example 38 Not in Syllabus - CBSE Exams 2021
Example 39 Not in Syllabus - CBSE Exams 2021
Example 40 Not in Syllabus - CBSE Exams 2021
Example 41 Important
Example 42 Important
Example 43 Important You are here
Example 44
Example 45 Important Not in Syllabus - CBSE Exams 2021
Example 46 Important
Example 47 Important
Example 48 Important
Example 49
Example 50
Example 51 Important
About the Author