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

1.	Show that the function f: N → N: f(x) = x2 is one-one into.
Answer» Consider N as the set of all natural numbers It is given that f: N → N: f(x) = x² Ɐ x ∈ N We know that f(x₁) = f(x₂) It can be written as x₁² = x₂² So we get x₁² – x₂² = 0 We get (x₁ – x₂) (x₁ + x₂) = 0 Here, (x₁ – x₂) = 0 where x₁ = x₂ Hence, f is one-one. Consider A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25} Here, f: A → B: f(x) = x² By substituting the values f(1) = 1² = 1 f(2) = 2² = 4 f(3) = 3² = 9 f(4) = 4² = 16 Hence, every moment in A has unique image in B. Here, ∃ 25 ∈ B which has no pre-image in A f is into.

Answer»

Consider N as the set of all natural numbers

It is given that f: N → N: f(x) = x² Ɐ x ∈ N

We know that

f(x₁) = f(x₂)

It can be written as

x₁² = x₂²

So we get

x₁² – x₂² = 0

We get

(x₁ – x₂) (x₁ + x₂) = 0

Here, (x₁ – x₂) = 0 where x₁ = x₂

Hence, f is one-one.

Consider A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}

Here, f: A → B: f(x) = x²

By substituting the values

f(1) = 1² = 1

f(2) = 2² = 4

f(3) = 3² = 9

f(4) = 4² = 16

Hence, every moment in A has unique image in B. Here, ∃ 25 ∈ B which has no pre-image in A f is into.

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