NCERT Class 12 · Ch 1 Exercise 1.2 · Q3
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 3
Prove that f:ℝ→ℝ, f(x)=⌊x⌋, is neither one-one nor onto.
💡 Concept Hints
- Find two distinct numbers in the same unit interval, then identify the range.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: Neither injective nor surjective.
- For instance ⌊1/4⌋=⌊3/4⌋=0 with distinct real inputs, so f is not injective.
- Every floor value is an integer, and every integer k is reached by input k; hence the range is exactly ℤ.
- Since 1/2∈ℝ∖ℤ is not reached, f is not onto the declared codomain ℝ.
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