Talk:Math 360, Fall 2021, Assignment 1

Revisions (Michael Reilly):

1. Revised definition of “containment”: containment only refers to relationships between sets, “membership” is the term discussed in class that applied to individual elements in a set.
2. Revised definition of “equality”: equality does not care about cardinality, because sets can be equal and have different cardinalities, if an element is repeated. Set equality is only concerned with whether the sets are subsets of each other (ie., all elements of each set are elements of the other set).
3. revised definition of “property”: it is important to be specific that a “property” must be a true or false statement, the original definition was ambiguous.
4. revised book problem (4): the instructions said specifically to list the elements of the set, the set was already described using a rule. Also, the rule that was given as an answer to this question said that all integers between -10.2 and 11.2 are included, and while that answer logically makes sense as there are integers between those values, typically integers are listed with integer bounds, so it would be $-10 \leq x \leq 11$.
5. revised proof of largest set: The proof was hard to follow, and I don’t think even proved what the question asked. The proof was a proof by contradiction, but I think it followed faulty logic. The basic structure was: “Proof of Proposition P. Assume P is false. P=false creates a contradiction, therefore P is false.” Luckily, there is a much simpler proof that is based on the definition of the Power Set, which I have put there instead.

More Changes

1. In my proof, I first assume “There is largest set is true, and S is this largest set”, then I prove that “there is a set larger than S”, therefore “S is not the largest set”, proved by contradiction that “There is a largest set S is not true”. I think it would work.
2. and about equality, I found at storyofmathematics.com that “Two sets are said to be equal if they contain the same elements and the same cardinality. This concept is known as Set Equality.” I can’t remember clearly, but have we talked about not considering multisets in class?

- We have not discussed them in class yet, no. The definition I put is the definition we were given in class, where $\{1,2\} \text{ and } \{1,2,2\}$ were said to be equivalent. Maybe that definition applies to “simplified” sets, where duplicates are omitted?

1. $\{1,2,2\}$ is a multi set, I remembered we talked about it in class. But I don’t remember if we should take it into consideration, or multi set is not considered in Abstract algebra? If we should, then I would agree that cardinalities don’t have to be the same.