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.
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Chapter 1 Class 12 Relation and Functions
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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.