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Question
Give an example of a function which is one-one but not onto ?
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Solution
which is one-one but not onto.
f: Z → Z given by f(x) = 3x + 2
Injectivity:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
f (x)= f(y)
⇒ 3x + 2 =3y + 2
⇒ 3x = 3y
⇒ x = y
⇒ f(x) = f(y) ⇒ x = y
So, f is one-one.
Surjectivity:
Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z(domain).
f(x) = y
⇒ 3x + 2 = y
⇒ 3x = y - 2
⇒ x= `(y - 2)/3`. It may not be in the domain (Z)
because if we take y = 3,
`x = (y - 2)/3 = (3-2)/3 = 1/3 ∉` domain Z.
So, for every element in the co domain there need not be any element in the domain such that f(x) = y.
Thus, f is not onto.
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