Texts: Problem 3 (23 pts.) Expected Time: 1-2 hours.
Similar to: Claims in Introduction to Functions, Tutorial 4 Problems 1 & 3.
Type of Practice: Construct Proof with Less Guidance.
This problem asks you to construct a correct and precise proof by contrapositive about sets and functions, with less guidance.
Claim: Let A, B, C be sets. If f: A → C is not a function, then either B ⊆ C or f: A → B is not a function. (Everything after the "then" is the consequent, to disambiguate double negations, etc. Be careful of the "or"! You can use predicate logic to help you here.)
b) 4 pts. Since this is an "if S then T" proof, you should assume S (the antecedent) and then want to show T (the consequent). Given your simplification of the antecedent in (a), and the 3 properties of a function from Problem 2, what assumptions does your antecedent allow you to make, formally, in terms of these properties, in terms of f, A, B, C?
c) 3 pts. Since this is an "if S then T" proof, you should assume S (the antecedent) and then want to show T (the consequent). Given your simplification of the consequent in (a), and the 3 properties of a function from Problem 2, what do you want to show, formally, in terms of these properties, in terms of f, A, B, C?
d) 13 pts. Fill in the table below with a proof by contrapositive of the claim. You must use algebraic notation in your mathematical statements column, and reasons from the approved list as your reasons. Recall that a proof by contrapositive has to start with rewriting the original implication in its contrapositive form, then it has to assume the left-hand side of the implication (the antecedent), then try to prove the right-hand side (the consequent). That is, you should start with the assumptions from b and use them (and other definitions, facts, reasons) to try to prove the propositions in (c).
There are 3 properties to a function, so if your proof is trying to prove something is a function it will need 3 separate sub-proofs, one per property, as you did in Problem 2.
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 a beneficial experience. The proof chatGPT gave me didn't understand the 3 properties and thus is not correct. Ultimately, you are responsible for putting whatever proof you have into the format used in this course.
We saw proofs about functions in the Introduction to Functions lecture, although since the properties of functions are not individually complex, it is more relevant to look at proofs by contrapositive from Tutorial 4 Problems 1 & 3, as well as our lecture on Proofs by Contrapositive. Understand these before attempting this; you can and should use those proofs as a basis for this one.
You should have between 6 and 10 rows in your table.
Hint: Don't forget to apply the problem-solving process. Try some examples first, and look at related results. If you are stuck for more than 20 minutes, please ask for help on Piazza.
