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Question
Show that the function f : N → N defined by f(x) = 2x – 1 is one-one but not onto
Solution
f : N → N
N = {1, 2, 3, 4, 5, …}
f(x) = 2x – 1
f(1) = 2(1) – 1 = 2 – 1 = 1
f(2) = 2(2) – 1 = 4 – 1 = 3
f(3) = 2(3) – 1 = 6 – 1 = 5
f(4) = 2(4) – 1 = 8 – 1 = 7
f(5) = 2(5) – 1 = 10 – 1 = 9
f = {(1, 1) (2, 3) (3, 5) (4, 7) (5, 9) …..}
(i) Different elements have different images. This function is one function.
(ii) Here Range is not equal to co-domain. This function, not an onto function.
∴ The given function is one-one but not an onto.
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