Show that the function f : N → N : f (x) = x2 is one-one and into.

1.	Show that the function f : N → N : f (x) = x2 is one-one and into.
Answer» 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 \(x\in N\), we can’t have negative values ∴ f(x) is one-one f(x) = x² Let f(x) = y such that \(y\in N\) ⇒ y = x² ^{\(\Rightarrow x=\sqrt{y}\)} If y = 2, as \(y\in N\) Then we will get the irrational value of x, but \(x\in N\) Hence f(x) is not into Hence Proved

Answer»

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 \(x\in N\), we can’t have negative values

∴ f(x) is one-one

f(x) = x²

Let f(x) = y such that \(y\in N\)

⇒ y = x²

^{\(\Rightarrow x=\sqrt{y}\)}

If y = 2, as \(y\in N\)

Then we will get the irrational value of x, but \(x\in N\)

Hence f(x) is not into

Hence Proved

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