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

Last updated at Jan. 28, 2020 by Teachoo

Last updated at Jan. 28, 2020 by Teachoo

Transcript

Example 12 Show that f : N โ N, given by f(x) = {โ(๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐@๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐)โค is both one-one and onto. Check one-one There can be 3 cases x1 & x2 both are odd x1 & x2 both are even x1 is odd & x2 is even If x1 & x2 are both odd f(x1) = x1 + 1 f(x2) = x2 + 1 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f(x1) = f(x2) x1 + 1 = x2 + 1 x1 = x2 If x1 & x2 are both are even f(x1) = x1 โ 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 โ 1 = x2 โ 1 x1 = x2 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 If x1 is odd and x2 is even f(x1) = x1 + 1 f(x2) = x2 โ 1 If f(x1) = f(x2) x1 + 1 = x2 โ 1 x2 โ x1 = 2 which is impossible as difference between even and odd number can never be even Hence, if f(x1) = f(x2) , Then x1 = x2 โด function f is one-one Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Check onto f(x) = {โ(๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐@๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐)โค Let f(x) = y , such that y โ N x = {โ(๐ฆโ1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐@๐ฆ+1, ๐๐ ๐ฆ ๐๐ ๐๐๐)โค If x is odd f(x) = x + 1 y = x + 1 y โ 1 = x x = y โ 1 If x is odd, y is even If x is even f(x) = x โ 1 y = x โ 1 y + 1 = x x = y + 1 If x is even, y is odd Hence, if y is a natural number, x will also be a natural number i.e. x โ N Thus, f is onto.

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 You are here

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

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

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.