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Let f : N to N : f(x) =2 x for all x...

Let f : N `to N : f(x) =2 x` for all `x in N`
Show that f is one -one and into.

Text Solution

Verified by Experts

Let `x_(1),x_(2) in N`, then
`f(x_(1))=f(x_(2))`
`implies 2x_(1)=2x_(2)impliesx_(1)=x_(2)`
`therefore` f is one-one
Let y = 2x, then `x = (y)/(2)`
Now, if we put y = 5, then `x=(5)/(2)cancelinN`.
This show that `5 in N` has no pre-image in N. So, f is into. Hence, f is one-one and into.
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