# Homework Help: Power set P(S) with symmetry difference.

1. Feb 29, 2012

### mikael27

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Feb 29, 2012

### Dick

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

3. Feb 29, 2012

### mikael27

its A Δ B not A Δ A.

I dont know how to start this

4. Feb 29, 2012

### Dick

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?

5. Feb 29, 2012

### mikael27

the identity is the empty set.

6. Mar 1, 2012

### Deveno

ok, now: what is A-A?

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

7. Mar 1, 2012

### mikael27

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

8. Mar 1, 2012

### 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 gk = e.

what is gk? 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 Ak = AΔAΔA....ΔA (k times).

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

9. Mar 1, 2012

### mikael27

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

10. Mar 1, 2012

### Dick

Once again. What is AΔA?

11. Mar 1, 2012

### mikael27

Its A^2

12. Mar 1, 2012

### Dick

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

13. Mar 1, 2012

### mikael27

just A and A ?

14. Mar 1, 2012

### Dick

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

15. Mar 1, 2012

### mikael27

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

16. Mar 1, 2012

### Dick

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.

17. Mar 1, 2012

### mikael27

18. Mar 1, 2012

### Dick

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}?

19. Mar 1, 2012

### mikael27

yes i know that thats 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

20. Mar 1, 2012

### HallsofIvy

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

21. Mar 1, 2012

### Dick

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?

22. Mar 1, 2012

### mikael27

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

23. Mar 1, 2012

### Dick

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?

24. Mar 1, 2012

### mikael27

i dont know

25. Mar 1, 2012

### mikael27

infinity order