Finding Functions There are a lot of functions out there with interesting properties. This…

Finding Functions There are a lot of functions out there with interesting properties. This question asks you to think about different possible functions and what they might look like.

i. Find a function f : ℕ → ℕ that is both injective and surjective. Prove it meets those criteria.

ii. Find a function g : ℕ → ℕ that is injective but not surjective. Prove it meets those criteria.

iii. Find a function h: ℕ → ℕ that is not injective but is surjective. Prove it meets those criteria.

iv. Find a function k : ℕ → ℕ that's neither injective nor surjective. Prove it meets those criteria.

v. Based on your answers to these problems, explain why if you have a function f : A → B and you know that |A| = |B|, you cannot necessarily say anything about whether f is injective, surjective, or bijective.

 

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