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Question 21 (Choice 2) - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by

Prove that the function f is surjective, where 𝑓: 𝑁 → 𝑁 such that

 f(n)={(n + 1)/2,  if n is odd n/2,  if n is even

 

Is the function injective? Justify your answer.

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Transcript

Question 21 (Choice 2) Prove that the function f is surjective, where 𝑓: 𝑁 → 𝑁 such that 𝑓(𝑛)={█((𝑛 + 1)/2, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑@&𝑛/2, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛)┤ Is the function injective? Justify your answer. Check onto (surjective) 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 When n is even y = 𝑛/2 2y = n n = 2y Hence for y is a natural number , n = 2y is also a natural number Thus, for every y ∈ N, there exists x ∈ N such that f(n) = y Hence, f is onto (surjective) Check one-one (injective) f (n) = {█((𝑛 + 1)/2 ", if n is odd" @𝑛/2 ", if n is even" )┤ for all n ∈ N. 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 (not injective)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.