Show that the function f : N → N : f (x) = x2 is one-one and into.
To prove: function is one-one and into
Given: f : N → N : f (x) = x2
Solution: We have,
f(x) = x2
For, f(x1) = f(x2)
⇒ x12 = x22
⇒ x1 = x2
Here we can’t consider x1 = -x2 as 
, we can’t have negative values
∴ f(x) is one-one
f(x) = x2
Let f(x) = y such that 
⇒ y = x2
If y = 2, as 
Then we will get the irrational value of x, but 
Hence f(x) is not into
Hence Proved
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