Solving Set Theory Homework: Sets, Tuples, etc

[Panphobia]

Homework Statement

So if a question asks you for a pair set, with some criteria, is it enough to just say S = {a,b} or do you need something extra? Also this is another question that is from my set theory class, if the question defines a function and to solve over the Real numbers, ex. 1-|x^2-1| = 0, I know it isn't as simple as it looks, I know we are supposed to give all the values of x satisfying this function in a set, but I think our professor is looking for some kind of proof, like using, "if a = b then |a| = |b|". So my question is, how do I integrate those if statements while solving for all values of x?

 

[haruspex]

Homework Statement

So if a question asks you for a pair set, with some criteria, is it enough to just say S = {a,b} or do you need something extra?

There is a simple constraint for elements of a set that does not apply to elements of a tuple.

Also this is another question that is from my set theory class, if the question defines a function and to solve over the Real numbers, ex. 1-|x^2-1| = 0, I know it isn't as simple as it looks, I know we are supposed to give all the values of x satisfying this function in a set, but I think our professor is looking for some kind of proof, like using, "if a = b then |a| = |b|". So my question is, how do I integrate those if statements while solving for all values of x?

There are three ways I can of for dealing with modulus signs.
1. Square
In this example you can write (x²-1)² = 1. Not useful here, though.
2. Consider each combination of the binary choices separately.
To illustrate that I need a more complicated example: 1-|x^2-1| = |x+3|
The four cases to be considered correspond to x²>1 versus x²<1, and x>-3 versus x<-3. Some combinations may turn out to be impossible.
3. Find all the critical values of x and divide the real line up into corresponding intervals. In my example, the critical values are x=-1, x=+1, x=-3, so the line is divided into four intervals.

Generally, method 3 scales better than method 2 (which grows cases exponentially).

 

[Panphobia]

Thanks, that helped, I figured out the solution set to all parts of that question.