Question:
Give an example of a map
(i) which is one-one but not onto
(ii) which is not one-one but onto
(iii) which is neither one-one nor onto.
Solution:
(i) Let f: N → N, be a mapping defined by f (x) = x2
For f (x1) = f (x2)
Then, x12 = x22
x1 = x2 (Since x1 + x2 = 0 is not possible)
Further ‘f’ is not onto, as for 1 ∈ N, there does not exist any x in N such that f (x) = 2x + 1.
(ii) Let f: R → [0, ∞), be a mapping defined by f(x) = |x|
Then, it’s clearly seen that f (x) is not one-one as f (2) = f (-2).
But |x| ≥ 0, so range is [0, ∞].
Therefore, f (x) is onto.
(iii) Let f: R → R, be a mapping defined by f (x) = x2
Then clearly f (x) is not one-one as f (1) = f (-1). Also range of f (x) is [0, ∞).
Therefore, f (x) is neither one-one nor onto.