A few homework problems (due Friday)

[Jamin2112]

Disclaimer: This is a college course of mine in which the homework is scrutinized by the professor and worth 30% of our grade. I've done all the homework problems by myself except these 2. Do not, in any way, tell me how to do it. It's okay to explain something that my attempt shows I'm fuzzy on, or to remind me of a certain fact of math reasoning that could lead me in the right direction. Just be wary. Here is a link to the assignment, just in case my writing transcript doesn't do it justice: http://www.math.washington.edu/~folland/hw4.pdf

Homework Statement

(I don't know how to make all the fancy symbols)

(A disunion B = A disunion C) and (A union B = A union C) equivalent to B = C

Homework Equations

?

The Attempt at a Solution

I'm thinking proof by contradiction

(A disunion B ≠ A disunion C) or (A union B ≠ A union C) equivalent to B ≠ C

It seems like common sense but I can't figure out how to explain it. "Pictures are not proofs," my professor says. I mean, obviously if the intersection of A and B is equal to the intersection of B and C then B=C, right?

Help me get on the right track here.

Homework Statement

Suppose that A is a subset of Z (integers). Write the following statements entirely in symbols using the quantifiers A (an upside down A) and E (a backwards E). Write out the negative of this statement in symbols.

There is a greatest number in the set A.

Give an example of a set A for which this statement is true. Give another example for which it is false.

Homework Equations

The upside down A means "for each" and the backwards E means "there exists a". Just in case you didn't know.

The Attempt at a Solution

So...

The part that has me stuck is the "There is a greatest number...". I can't figure out how that would be written in symbols. It would come in the predicate, no doubt. Something like {.... : a_k > a_k+1 }. Nothing in the book speaks of writing a proof for "there is a greatest number". I'm even sure I understand what that means. Does it mean we have a set that goes something like 1,2,3,...,n and n is the greatest number? If so, what sort of predicate says that?

 

[HallsofIvy]

I'm not clear on what you mean by "disunion". I am going to assume you might mean simply $A\cap B$, the intersection, but it also occurred to me that you might mean the "symmetric difference", A\B, the set of all members of A that are NOT in B.

To prove "X equivalent to Y", first prove "if X then Y" and prove "if Y then X".

To prove "A= B" where A and B are sets, first prove "$A\subset B$", then prove "$B\subset A$".

To prove "$A\subset B$" start "if x is in A" and use whatever properties you know of A and B to show "then x is in B".

To prove "A disunion B = A disunion C) and (A union B = A union C) equivalent to B = C" first prove "if A disunion B = A disunion C) and (A union B = A union C) then B = C"
To prove that, basically, you want to use "A disunion B = A disunion C) and (A union B = A union C)" to prove B= C so you start by saying "if x is in B". Now what can you say about x that will lead to proving it is in C? Knowing that x is in B tells you that it is in $A\cup B$ and then because you know $A\cup B= A\cup C$, it follows that x is in $A\cup C$. From that, either x is in C or it is in A.
If it is in C we are done, so focus on the case that it is in A. If x is in A, since we already know it is in B, x is in $A\cap B$ and we know that $A\cap B= A\cap C$, we know that x is in $A\cap C$. But any member of $A\cap C$ is in C so either way we know that x is in C.

 

[ystael]

For the second problem: If x is the greatest member of A, what is the relationship between x and any other member y of A? Does that relationship capture everything about "being the greatest member of A", or is there more to it?