Question

For each of the functions below, indicate whether the function is onto, one-to-one, neither orboth. If the function is not onto or not one-to-one, give an example showing why.

(a)f1:Z→Z.f1(x) =x33
(b)f2:Z→Z.f2(x) =bx3c+ 23
(c)f3:Z×Z→Z×Z.f3(x, y) = (x+ 1,2y)3
(d)f4:Z+×Z+→Z+.f4(x, y) = 2x+y−1

Answer 1

`f_(1) :Z \rightarrow Z` is one to one mapping, it is not onto mapping

Explanation:

`f_(1) :Z \rightarrow Z\\ f_(1) (x) = x^(3)`

f₁(x) is one to one mapping

Let
`x, y \epsilon Z`

f₁(x) = f₁(y):

x₁³ = y₁³

f₁(x) is not onto mapping

Example: If f₁(x) = 7,

x₁³ = 7

`x_(1) = \sqrt[3]{7}`

x₁ is not an element of Z

`f_(1) :Z \rightarrow Z` is one to one mapping, it is not onto mapping