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Example 12 - Show that f is both one-one onto f(x) = {x + 1 - To prove injective/ surjective/ bijective (one-one & onto)

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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Example 12 Show that f : N โ†’ N, given by f(x) = ๐‘ฅ+1 , ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฅโˆ’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 โˆด function f is one-one Check onto f(x) = ๐‘ฅ+1 , ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฎ๐‘ฅโˆ’1, ๐‘–๐‘“ ๐‘ฅ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฏ๏ทฏ Let f(x) = y , such that y โˆˆ N x = ๐‘ฆโˆ’1 , ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›๏ทฎ๐‘ฆ+1, ๐‘–๐‘“ ๐‘ฆ ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘๏ทฏ๏ทฏ 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 is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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