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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
  3. Examples

    Example 1

    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 You are here

    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

    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


Transcript

Example 27 Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Show that f, g and gof are invertible. Find out f –1, g –1 and (gof) –1 and show that (gof) –1 = f –1 o g –1. Checking for f f : {1, 2, 3} → {a, b, c} f (1) = a, f (2) = b, f (3) = c, f is invertible if it is one-one and onto Check one-one Since, all elements have unique image f is one-one Check onto Since, every image has a unique pre-image, ∴ f is onto Since f is one-one and onto, f is invertible Now, f = {(1, a), (2, b) , (3, c)} So, f-1 = {(a, 1), (b, 2), (c, 3)} Checking for g g : {a, b, c} → {apple , ball , cat} g(a) = apple, g(b) = ball , g(c) = cat, g is invertible if it is one-one and onto Check one-one Since, all elements have unique image g is one-one Check onto Since, every image has a unique pre-image g is onto Since g is one-one and onto g is invertible So, g = {(a, apple) , (b, ball) , (c, cat)} ∴ g–1 = {(apple, a), (ball, b), (cat, c)} Checking for gof So, gof will be gof = { (1, apple) , (2, ball) , (3, cat) } gof is invertible if it is one-one and onto Check one-one Since, all elements have unique image gof is one-one Check onto Since, every image has a unique pre-image gof is onto Since gof is one-one and onto gof is invertible So, gof = { (1, apple) , (2, ball) , (3, cat) } ∴ (gof)–1 = {(apple, 1), (ball, 2), (cat, 3)} We need to show that (gof) –1 = f –1 o g –1 Finding f –1 o g –1 f-1 = {(a, 1), (b, 2), (c, 3)} g–1 = {(apple, a), (ball, b), (cat, c)} Hence, (gof)–1 = {(apple, 1), (ball, 2), (cat, 3)} f –1 o g –1= {(apple, 1), (ball, 2), (cat, 3)} Thus, (gof) –1 = f –1 o g –1 Hence proved

Examples

Example 1

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 You are here

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

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

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.