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  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) = {โ–ˆ(๐‘ฅ+1 , ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘@๐‘ฅโˆ’1, ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›)โ”ค 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 โˆด 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–ˆ(๐‘ฅ+1 , ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘@๐‘ฅโˆ’1, ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›)โ”ค Let f(x) = y , such that y โˆˆ N x = {โ–ˆ(๐‘ฆโˆ’1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›@๐‘ฆ+1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘)โ”ค 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 โˆˆ N Thus, f is onto.

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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.