Example 12 - Chapter 1 Class 12 Relation and Functions
Last updated at Nov. 5, 2018 by Teachoo
Transcript
Example 12 Show that f : N โ N, given by f(x) = ๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐๏ทฎ๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ is both one-one and onto. Check one-one There can be 3 cases โข x1 & x2 both are odd โข x1 & x2 both are even โข x1 is odd & x2 is even If x1 & x2 are both odd f(x1) = x1 + 1 f(x2) = x2 + 1 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 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) , x1 = x2 โด function f is one-one Check onto f(x) = ๐ฅ+1 , ๐๐ ๐ฅ ๐๐ ๐๐๐๏ทฎ๐ฅโ1, ๐๐ ๐ฅ ๐๐ ๐๐ฃ๐๐๏ทฏ๏ทฏ Let f(x) = y , such that y โ N x = ๐ฆโ1 , ๐๐ ๐ฆ ๐๐ ๐๐ฃ๐๐๏ทฎ๐ฆ+1, ๐๐ ๐ฆ ๐๐ ๐๐๐๏ทฏ๏ทฏ 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
Example 7
Example 8
Example 9
Example 10
Example 11 Important
Example 12 Important You are here
Example 13 Important
Example 14
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
Example 28 Important
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
Example 42 Important
Example 43
Example 44
Example 45 Important
Example 46 Important
Example 47 Important
Example 48 Important
Example 49
Example 50
Example 51
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.