CBSE Class 12 Sample Paper for 2023 Boards
You are here
CBSE Class 12 Sample Paper for 2023 Boards
Last updated at Dec. 13, 2024 by Teachoo
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)
