Question

Discrete math question

Prove f(n) = n² - 1 is one-to-one (injective)

Answer 1

To prove that the function f(n) = n² - 1 is one-to-one (injective), we can show that different inputs result in different outputs.

Step-by-step explanation:

To prove that the function f(n) = n² - 1 is one-to-one (injective), we need to show that different inputs result in different outputs. We can do this by using algebraic manipulation.

1. Assume that f(n₁) = f(n₂), where n₁ and n₂ are any two different numbers.
2. Substituting the function expression, we have n₁² - 1 = n₂² - 1.
3. Simplifying the equation, we get n₁² = n₂².
4. Taking the square root of both sides, we have |n₁| = |n₂|.
5. Since n₁ and n₂ are different numbers, their absolute values must also be different.
6. Therefore, f(n₁) ≠ f(n₂), which means the function f(n) = n² - 1 is one-to-one (injective).