Power set P(S) with symmetry difference.

[mikael27]

Homework Statement

Determine the orders of all the elements of the power set P(S) of a set S with symmetric difference Δ.

Homework Equations

The Attempt at a Solution

If A,b are two elements of the power set
the symmetric difference is

AΔB = (A-B) U (B - A)

How are we going to find the order of the elements?

 

[Dick]

What's AΔA? That's A^2 with your group operation.

 

[mikael27]

its A Δ B not A Δ A.

I don't know how to start this

 

[Dick]

its A Δ B not A Δ A.

I don't know how to start this

You would if you paid some attention to me. The order of A is the number of times you have to take the symmetric difference of A with itself to get the group identity. What's the identity of your group?

 

[mikael27]

the identity is the empty set.

 

[Deveno]

ok, now: what is A-A?

and therefore, what is (A-A) U (A-A) = AΔA?

 

[mikael27]

I also know that AΔB = (A-B) U (B - A)= (B-A) U (A - B)=BΔA

 

[Deveno]

well, that's very nice, but:

in a group, the order of an element g is the smallest possible positive integer k such that g^k = e.

what is g^k? it's g*g*g*...*g (k times).

so when calculating order of an element, you don't need to look at "other elements".

in the group you are considering, we are using "A" instead of "g", and "Δ" instead of "*".

so A^k = AΔAΔA...ΔA (k times).

there's no need to bring up "B".

 

[mikael27]

To the answer is that we have one element A and the order of A is k?

 

[Dick]

To the answer is that we have one element A and the order of A is k?

Once again. What is AΔA?

 

[mikael27]

Its A^2

 

[Dick]

Its A^2

Ok, yes, but what is that? What elements are in that set?

 

[mikael27]

just A and A ?

 

[Dick]

just A and A ?

Use your definition! AΔA = (A-A) U (A-A). What is that? It's a very simple set.

 

[mikael27]

it says that the A-A union A-A. On sets if A-A=0 it means it is zero

 

[Dick]

it says that the A-A union A-A. On sets if A-A=0 it means it is zero

A-A isn't 0, if you mean the number zero. Do you know what A-A means? It's a set, not a number.

 

[mikael27]

can you please explain me

 

[Dick]

can you please explain me

A-B means the set of all the elements of A that are not in B. What is {1,2,3,4}-{5,4,2,7}? Now what is {1,2,3,4}-{1,2,3,4}?

 

[mikael27]

yes i know that that's why i told you 0. There are no elements left

{1,2,3,4}-{5,4,2,7} 1,3

{1,2,3,4}-{1,2,3,4} no elements

 

[HallsofIvy]

And what do you call a set that contains no elements? That's what Dick is asking.

 

[Dick]

yes i know that that's why i told you 0. There are no elements left

{1,2,3,4}-{5,4,2,7} 1,3

{1,2,3,4}-{1,2,3,4} no elements

That's good. 0 isn't a good answer. 'empty set' or {} or ∅ is the way to say that. So A^2={}. What does that tell you about the order of A?

 

[mikael27]

so A^2=empty set. no elements from -infinity to + infinity

 

[Dick]

so A^2=empty set. no elements from -infinity to + infinity

That has nothing to do with the question I asked you. What does A^2={} tell you about the order of A? What's the identity of P(A)? Try to focus, ok?

 

[mikael27]

i don't know

 

[mikael27]

infinity order

 

[HallsofIvy]

You said, back in response #5, that "the identity is the empty set."''

Of course, "is" is symmetric! "The empty set is the identity". So, calling the identity "e", $A\Delta A= \phi$ is the same as $A\Delta A= e$.

Now, what does that tell you about the "order" of A?zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

 

[Dick]

infinity order

I want to you reread Deveno's explanation in post 8. Look up what the 'order of a element in a group' means in your textbook or on line. Figure out why P(A) is a group under Δ. Then think about it for a while, come back here and explain to me what it means in your own words. Then if you still don't know we can continue.

 

[micromass]

Mikael, you're giving me the impression that you don't really want to think about the question. This forum is not for spoonfeeding you or giving you the answer without effort.
If you continue to do no effort then this thread will be locked.