Ex 1.2 , 9 - Let f(n) = {n+1/2, if n is odd n2, if n is even - Ex 1.2

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  1. Chapter 1 Class 12 Relation and Functions
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Ex 1.2 , 9 Let f: N → N be defined by f (n) = 𝑛 + 1﷮2﷯, if n is odd﷮ 𝑛﷮2﷯, if n is even﷯﷯ for all n ∈ N. State whether the function f is bijective. Justify your answer. f (n) = 𝑛 + 1﷮2﷯, if n is odd﷮ 𝑛﷮2﷯, if n is even﷯﷯ for all n ∈ N. Check one-one f(1) = 1 + 1﷮2﷯ = 2﷮2﷯ = 1 f(2) = 2﷮2﷯ = 1 Since, f(1) = f(2) but 1 ≠ 2 Both f(1) & f(2) have same image 1 ∴ f is not one-one Check onto f (n) = 𝑛 + 1﷮2﷯, if n is odd﷮ 𝑛﷮2﷯, if n is even﷯﷯ for all n ∈ N Let f(x) = y , such that y ∈ N When n is odd y = 𝑛 + 1﷮2﷯ ⇒ 2y = n + 1 ⇒ 2y – 1 = n ⇒ n = 2y – 1 Hence, for y is a natural number , n = 2y – 1 is also a natural number Thus, f is onto

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