Class 12 Mathematics - Chapter Relations and Functions NCERT Solutions | Prove that the Greatest Integer Fu
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
f: R → R is given by,
f(x) = [x]
It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1.
∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9.
∴ f is not one-one.
Now, consider 0.7 ∈ R.
It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7.
∴ f is not onto.
Hence, the greatest integer function is neither one-one nor onto.
More Questions From Class 12 Mathematics - Chapter Relations and Functions
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(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
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(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
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(b) R = {(x, y): x and y live in the same locality}
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(C) f is one-one but not onto
(D) f is neither one-one nor onto.
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