NCERT Class 12 · Ch 1 Exercise 1.2 · Q9
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 9
For ℕ={1,2,…}, define f(n)=(n+1)/2 when n is odd and f(n)=n/2 when n is even. Is f:ℕ→ℕ bijective?
💡 Concept Hints
- Compute the first few values, then use the even branch to reach an arbitrary natural target.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: Onto but not one-one; not bijective.
- The first two inputs give f(1)=1 and f(2)=1, so f is not injective.
- For arbitrary m∈ℕ, choose n=2m, an even natural number. Then f(n)=n/2=m, so f is onto.
- Every m also has preimage 2m−1; the two-to-one pattern confirms that f is not bijective.
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