SETS DEFINED BY SPECIFICATION: Given a set X and a sentence P(x) that is either true or false whenever x is any particular element of X, there is a set whose elements are precisely those x ∈ X for which P(x) is true, denoted by {x ∈ X : P(x)}.
Does this mean that whenever the function P(x) is true, then x is an element of X, and when P(x) is false, then x is not an element of X?

I'm confused because the wording says that "...a sentence P(x) that is either true or false whenever x is any particular element of X..." which leads me to believe that whether P(x) is either true or false, then it is still an element of set X.

Or is it saying that there is a set within X in which P(x) is true, and there is also another set within X in which P(x) is false?

2. Re: General question about sets

Hmm, I think the quote is precise and clear, but I am trying to understand your logic and what exctly you don't understand.

Originally Posted by PhizKid
Does this mean that whenever the function P(x) is true, then x is an element of X, and when P(x) is false, then x is not an element of X?
No, the quote does not say this.

Originally Posted by PhizKid
I'm confused because the wording says that "...a sentence P(x) that is either true or false whenever x is any particular element of X..." which leads me to believe that whether P(x) is either true or false, then it is still an element of set X.
What does "it" refer to, i.e., what is still an element of X: P(x) or x? And what exactly is x in this context?

Originally Posted by PhizKid
Or is it saying that there is a set within X in which P(x) is true, and there is also another set within X in which P(x) is false?
What do you mean by saying that P(x) is true "in" a set within X (i.e., a subset of X)? P(x) can only be true or false for a specific element x.

Let X be the set of former wife-beaters. Then whenever x is an element of X, the statement "x has stopped beating his wife" is either true or false. Note that this statement may be neither true nor false if x is not an element of X, for example, when x is a computer or when x is a man who has never beaten his wife.

3. Re: General question about sets

Originally Posted by PhizKid
Does this mean that whenever the function P(x) is true, then x is an element of X, and when P(x) is false, then x is not an element of X?
I'm confused because the wording says that "...a sentence P(x) that is either true or false whenever x is any particular element of X..." which leads me to believe that whether P(x) is either true or false, then it is still an element of set X.
Or is it saying that there is a set within X in which P(x) is true, and there is also another set within X in which P(x) is false?
Take the example $X=\mathbb{R}$, the real numbers. And $P(x)$ to be " $x^2-4\le 0.$
Clearly $S=\{x:P(x)\text{ is true}\}=[-2,2]$. Now $3\notin S$ but $3\in X$.
Now which do you think it means?

4. Re: General question about sets

P(x) is not a function, it is a proposition, that is to say it is a statement about a variable x, that may be true or false, as x varies over X (ok, you could consider it a boolean function).

for example, if X = {the set of all integers), then one such P(x) could be:

"x is an even number".

this means we can either specify the even integers by:

{the set of even integers}, OR:

{x in Z: x is even}, that is, the subset of Z for which P(x) is true. this is especially convenient for sets for which there are too many elements to even try to list, such as sets of real numbers.

for example we can list {all real numbers that are integers and greater than 0 and less than 2} as {1}. but listing {all real numbers that are less than 3} is not feasible.

the reason this "works" is due to the axiom of extensionality:

"For all A, and for all B: If for all C, C is an element of A if and only if C is an element of B, then A = B".

this lets us decide if two sets are equal, by comparing their members. so we can, for example, define a set A, by defining a predicate P(B). there is one caution: B should not mention A. for example:

K = the set of all elements of K

is a "bad definition", as the proposition P(x) = x is in K, mentions K itself (we haven't really "defined" anything. in other words, we don't want P(B) to mention A, to prevent "circular definitions", or, even worse, "self-contradictory" ones (to paraphrase Groucho Marx: i only belong to clubs that won't accept me as a member)).

it may come as a surprise to you that not all sets can be defined this way. the problem doesn't lie with the construction itself {x in X: P(x) is true}, but rather from the fact that for some propositions P(x), we cannot be sure P(x) is either true or false for every x (such propositions are called undecidable, and it is a famous theorem that in every logical system of "sufficient power", undecidable propositions exist).

fortunately, most propositions actually used, ARE decidable: for example, the set {x in Z: x is even}, is completely well-defined, because of any given integer x, we can determine if it is even by dividing it by 2. such a division will terminate in a finite number of steps, and give either a remainder of 0 (x is even is true), or 1 (x is even is false). if you stop and think about it, though, you can devise sets whose well-defined-ness is murky, at best. for example:

A = {The smallest positive integer not definable in under eleven words}

if A = {x}, then we have just defined x in ten words. so then A ≠ {x} = A, which is a most unpleasant state of affairs.

because of things of this nature, some mathematicians only accept "certain" P(x), which can be demonstrated to be either true or false. in other words, they reject the notion that ANY statement P(x) must be either true, or false, with no other alternative. for these people, the sets they have available to them is a smaller mathematical universe than those who are willing to accept that there is only true and false with no "in-between". fortunately for you, in actual practice, you will not often encounter such delicate questions, as most sets you will encounter are deemed acceptable by all.