Recent content by setsofvectors

I give up - I don't know what "x" I am talking about. all I know is that x is an element of S. I am not sure about the new symbol I might have introduced(?). Sounds like you know that specific "x" you're asking about. Can you tell me what it is, because I don't know?

I am not sure what you're asking. Do you need me to show a specific function that constructs C from certain x in S?

x is an element in S?

I made yet another mistake, but it looks like it just dawned on me why the statement "x is either in f(x) or x is not in f(x)." is included in the proof. Depending on the mapping function, some subsets of S don't contain x, some do. If we can prove that there's at least one subset of S that...

Let me see if I got a little closer to understanding the proof. First, the proof is either true or false. Roughly speaking, if it's false there's onto relation and x is in f(x). If it's true, there's no onto relation and x is not in f(x). The rest of the proof deals with showing x is...

Ok, in what situations can you invoke LEM? Thanks.

It comes from my raging desire to link sentences into logically consistent chains. So that turns out to be wrong. Let's scratch the part you don't understand. Does the derivation of "x is in f(x) or x is not in f(x)" look ok now?

Does the post #12 look in\correct to you?

It's not true that a statement is both true and false. -(p and -p) p or -(p and -p) -p ------------------------------------------------ -(cold and hot) cold or -(cold and hot) hot ------------------------------------------------------- We can also flip them...

Thank you. "The law of excluded middle states that for any proposition, either that proposition is true, or its negation is true." So, "x is an element of C" has to be the proposition according to the law, correct? So then, this proposition has to be linked to "Then there exists an x in S...

"Where does "Either x is in C or x is not in C' come from?" Assume that f maps S onto P(S). -> Then there exists an x in S such that f(x) = C.-> Either x ∈ C or x ∉ C -> contradiction which makes the assumption invalid. We start with an assumption and keep deriving logically linked sentences...

Do you mean S can equal C? Like this below?: Say, S = {1, 2}. Define functions as f(1) = {2}, f(2) = {1}. So, C = {1} and C = {2}. {1} U {2} = {1, 2}? Something like that? Where does "Either x is in C or x is not in C' come from?

You said "C can for example be the empty set or the whole space...". That's what I meant when I said both of those sets won't contain any x from S. Assumption goes f(x) = C, where x is in S. So we can suppose x is in the set C and the set C is in P(S).

P(S) contains f(x)'s. Depending on a function, there will be sets in P(S) where x is not in f(x). So there's a need to define C so that we can refer to such elements of P(S). ? Say, you're the first person in history who've come up with this proof. What would you have done first- define C or...

Hi, tiny-tim :smile: But neither of those sets contain x anyway? Isnt "Then there exists an x in S such that f(x) = C." saying suppose there's x in C which is in P(S)? Thanks.