1. Chapter 1 Class 12 Relation and Functions
2. Serial order wise

Transcript

Example 12 Show that f : N โ N, given by f(x) = ๐ฅ+1 , ๐๐ ๐ฅ ๐๐  ๐๐๐๏ทฎ๐ฅโ1, ๐๐ ๐ฅ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ is both one-one and onto. Check one-one There can be 3 cases โข x1 & x2 both are odd โข x1 & x2 both are even โข 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 โด function f is one-one Check onto f(x) = ๐ฅ+1 , ๐๐ ๐ฅ ๐๐  ๐๐๐๏ทฎ๐ฅโ1, ๐๐ ๐ฅ ๐๐  ๐๐ฃ๐๐๏ทฏ๏ทฏ Let f(x) = y , such that y โ N x = ๐ฆโ1 , ๐๐ ๐ฆ ๐๐  ๐๐ฃ๐๐๏ทฎ๐ฆ+1, ๐๐ ๐ฆ ๐๐  ๐๐๐๏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.