Question

Determine whether each of these functions from Z to Z is injective, surjective, bijective or none of these.

a) f(n) = n+1

b) f(n) = 2n

c) f(n) = 3n 2 - 1

d) f(n) = n 3

e) f(n) = floor(n/2)

Answer 1

In this question, we analyze functions to determine if they are injective, surjective, bijective, or none of these. We use the definitions of injective, surjective, and bijective to evaluate each function. The functions are analyzed and categorized accordingly.

Step-by-step explanation:

To determine whether each function is injective, surjective, bijective, or neither, we need to understand the definitions of these terms:

• Injective: A function is injective if each element in the domain maps to a unique element in the codomain.
• Surjective: A function is surjective if every element in the codomain has a corresponding element in the domain that maps to it.
• Bijective: A function is bijective if it is both injective and surjective, meaning each element in the domain maps to a unique element in the codomain, and every element in the codomain has a corresponding element in the domain.

Using these definitions, we can analyze each function:

• (a) Function f(n) = n+1 is injective, as each element in the domain corresponds to a unique element in the codomain.
• (b) Function f(n) = 2n is injective and surjective, as it satisfies both conditions.
• (c) Function f(n) = 3n^2 - 1 is neither injective nor surjective, as it fails to satisfy either condition.
• (d) Function f(n) = n^3 is injective, as each element in the domain corresponds to a unique element in the codomain.
• (e) Function f(n) = floor(n/2) is neither injective nor surjective, as it fails to satisfy either condition.