Example 27 - Let f(1) = a, f(2) = b, f(3) = c, g(a) = apple

Example 27 - Chapter 1 Class 12 Relation and Functions - Part 2
Example 27 - Chapter 1 Class 12 Relation and Functions - Part 3 Example 27 - Chapter 1 Class 12 Relation and Functions - Part 4 Example 27 - Chapter 1 Class 12 Relation and Functions - Part 5 Example 27 - Chapter 1 Class 12 Relation and Functions - Part 6

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Question 10 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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.