Determine a subset for the function of X -> P(X)

[hocuspocus102]

Homework Statement

Suppose X is a non-empty set and f:X->power set of X (ie P(X)) is defined f(x) = X/x. Consider the subset Y_f = {x in X such that x not in f(x)} of X. Determine Y_f for the particular f we have just described.

Homework Equations

The Attempt at a Solution

Any suggestions would be helpful because I really don't know where to start on this problem...

 

[Dick]

This really isn't a profound question. It looks like it's just testing if you can read the notation. Pick X={1,2,3}. What's f(1)?

 

[hocuspocus102]

f(1) in that case would be {2,3} right? but how does that translate to Y_f?

 

[Dick]

f(1) in that case would be {2,3} right? but how does that translate to Y_f?

Right. Is 1 in Y_f?

 

[hocuspocus102]

no, so is Y_f just X/x ?

 

[Dick]

no, so is Y_f just X/x ?

Read the definition of Y_f again. Aloud. I don't want to know what Y_f is yet, I just want to know if 1 is in it. No is the wrong answer. Why?

 

[hocuspocus102]

no 1 isn't in it because it's x in X such that x not in f(x) and 1 is not in f(1)

 

[Dick]

no 1 isn't in it because it's x in X such that x not in f(x) and 1 is not in f(1)

If 1 is not in f(1), doesn't that mean 1 IS in Y_f?

 

[hocuspocus102]

I get that for f(1), 2 is in Y_f and 3 is in Y_f but 1 is not.
for f(2), 1 is in Y_f and 3 is in Y_f but 2 is not.
so for f(n) on a set of size t Y_f will have all numbers in the set 1 through t not including n in it. is that how you'd say that or is there a more explicit formula?

 

[Dick]

I get that for f(1), 2 is in Y_f and 3 is in Y_f but 1 is not.
for f(2), 1 is in Y_f and 3 is in Y_f but 2 is not.
so for f(n) on a set of size t Y_f will have all numbers in the set 1 through t not including n in it. is that how you'd say that or is there a more explicit formula?

Ach. I think you are getting this wrong. 1 ISN'T in f(1). Doesn't that tell you 1 IS in Y_f? Or did you state the question wrong?

 

[hocuspocus102]

I don't know I thought f just took n to a set with all elements of X not including n because Y_f is x not in f(x)

 

[Dick]

I don't know I thought f just took n to a set with all elements of X not including n because Y_f is x not in f(x)

That's really confusing. Let's go back. If X={1,2,3} is 1 in Y_f? I don't think we really ever got that one right.

 

[hocuspocus102]

no it's not

 

[Dick]

no it's not

Yes, it is. Yf = {x in X such that x not in f(x)}. 1 is in X but it's NOT in f(1)={2,3}. So 1 IS in Y_f.

 

[hocuspocus102]

ohhhh. ok so Y_f is whatever value you plug in for x?

 

[Dick]

ohhhh. ok so Y_f is whatever value you plug in for x?

Hmm. What Y_f is actually doesn't depend on what x is. x is what's called a dummy variable. Work it out for X={1,2,3} and tell me what members of X are in Y_f.

 

[hocuspocus102]

{1,2,3} are in Y_f because f(1) implies 1 is in it, f(2) implies 2 is in it and f(3) implies 3 is in it right?

 

[Dick]

{1,2,3} are in Y_f because f(1) implies 1 is in it, f(2) implies 2 is in it and f(3) implies 3 is in it right?

Right. So Y_f=X={1,2,3}. Does the same thing work for every set?

 

[hocuspocus102]

it should

 

[Dick]

it should

It does. x is never in X/{x}.

 

[hocuspocus102]

so Y_f is just X?

 

[Dick]

so Y_f is just X?

What do YOU think? That's a lot more important than what I think.

 

[hocuspocus102]

that's what I think

 

[Dick]

that's what I think

I agree.

 

[hocuspocus102]

ok. Thank you very much!