Problem 2 (18 pts.) Expected Time: 1-2 hours. Similar to: Claims in Introduction to Functions, Tutorial 1. Type of Practice: Read, Parse and Deconstruct Given Proofs. This problem asks you to critically read a chatGPT proof and improve upon it. Then it asks you to construct some simple direct proofs, in the format used in this course (e.g. "column format".) Claim. Let f : 2" -> 2" be defined by f(X) = {x : re X and x is even}. Then f is a function. (2" is just alternate notation for P(Z), the powerset of Z.) Recall the definition: g : A - B is a function from domain A to codomain B if and only if (1) for each a E A, g(a) is computable/defined, (2) for each a E A, g(a) does not produce two different outputs, and (3) for each a E A, g(a) E B. Note. These are written as general properties that apply to any function g, domain A, and codomain B. When you apply these properties to this problem's particular function, domain, and codomain, you should make appropriate changes. Moreover, recall that by convention, any variable representing a set should be an uppercase letter. Therefore, when you apply these general properties to an element of this domain, you must use an uppercase variables instead of lowercase a. (a) (2 pts.) Let 2Z := {2c : c E Z}, that is, 2Z is the set of all even integers. Write f(X) algebraically using only set operations, X, and 2Z. Briefly explain (one reason suffices.) You shouldn't need chatGPT for this. (b) (4 pts.) Fill out the following table with a short direct proof that f has property (1). Use your formal definition from (a) for the most precise proof. Use the column format. You should have 2-4 steps after you state property (1) in terms of this problem's g and A. You can use chatGPT but since it doesn't really understand my properties, its answers will likely just be misleading and time wasting. You've been warned! Mathematical Statement Reason this Statement is True (From the Approved List)(c) (4 pts.) I asked chatGPT to show that f has property' 2, and it started out strong (see below) although in a different way than we do in lecture. It rephrases property 2 in a more precise wayas an if-then statementthen states how it plans to prove that if-then statement by assuming the antecedent, then proving the consequent. The rest of the proof (omitted), however, was more complex than necessary. Use this start to ll in the following table with a direct proof that f has property 2. Use your formal denition from (21). Use the column format. You should have 2-4 steps after you state property (2) more precisely and in term of this problem's g and A. You may use (and cite) chatGPT to try to ll in the rest of the proof, but be wary that for me it had a whole paragraph arguing why if A = B I could substitute A for B (which, is a one-liner: \"by substitution\" .) (e) (3 pts.) Consider changing the codomain to be subsets of even integers, i.e. f : 23 r 223. Is f still a function? (Recall that only property 3 relies on the codomain, so you can focus on that.) If you believe not, use a disproof by counter-example to show that property 3 doesn't hold with this new codomain. If you believe it remains a function, use a short direct proof quite similar to (d) where you prove it has property 3. Recall that a disproof by counter-example must: 0 State the counter-example (e.g. specic sets A, B and function f that satisfy the con- straints); 0 Apply the result to these values to get a proposition (apply the function for us); a Show why this proposition is false (this is typically straight-forward or obvious.) A direct proof would follow the same structure as given in (d). (d) (5 pts.) Fill out the following table with a direct proof that f has property 3. Use your formal definition from (a), and use formal set notation in your "mathematical statements" column, and reasons from the approved list as your reasons (i.e. column format.) You are permitted to use (then cite) chatGPT; however, I encourage you to practice solving the problem on your own as much as possible as this is as beneficial experience. Ultimately, you are responsible for putting whatever proof you have into the format used in this course