Question

Show that the function f : N → N defined by f(x) = 2x – 1 is one-one but not onto

Answer 1

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.