Subscribe to our Youtube Channel - https://you.tube/teachoo
Example 12 - Chapter 1 Class 12 Relation and Functions
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
Example 9
Example 10
Example 11 Important
Example 12 Important You are here
Example 13 Important
Example 14 Important
Example 15
Example 16
Example 17
Example 18 Important
Example 19 Important
Example 20
Example 21
Example 22
Example 23 Important
Example 24
Example 25 Important
Example 26
Example 27 Important
Example 28
Example 29
Example 30
Example 31
Example 32
Example 33
Example 34
Example 35
Example 36
Example 37
Example 38
Example 39
Example 40
Example 41 Important
Example 42 Important
Example 43 Important
Example 44
Example 45 Important
Example 46 Important
Example 47 Important
Example 48 Important
Example 49
Example 50
Example 51 Important
Chapter 1 Class 12 Relation and Functions
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 9 years. He provides courses for Maths and Science at Teachoo.