Symmetric difference in sets

  • Thread starter Uiiop
  • Start date
  • #1
15
0

Homework Statement



There is a symmetric difference in sets X & Y, X Y is defined to be the sets of elements that are either X or Y but not both
Prove that for any sets X,Y & Z that
(X[itex]\oplus[/itex]Y)[itex]\oplus[/itex](Y[itex]\oplus[/itex]Z) = X[itex]\oplus[/itex]Z

Homework Equations



[itex]\oplus[/itex] = symmetic difference

The Attempt at a Solution


i can see it in the venn diagams, but im not good at converting what i see into set statements this is my attempt in words

the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C because the symmetric difference of two repeated symmetric differences is the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed.

i dont have a clue how to show that properly in subsets etc that why i need some help
 

Answers and Replies

  • #3
22,129
3,298
How did you define [itex]X\oplus Y[/itex] in symbols??

You can split the proof up in this parts:
- Associativity: show that [itex]X\oplus (Y\oplus Z)=(X\oplus Y)\oplus Z[/itex]
- Show that [itex]Y\oplus Y=\emptyset[/itex]
- Show that [itex]Y\oplus \emptyset=Y[/itex]

(this actually implies that [itex]\oplus[/itex] forms a group operation). These three things together imply what you want to show, do you see that?
 
  • #4
15
0
ok here is my revised attempt at the answer with your help:

In order to prove that (X⊕Y)⊕(Y⊕Z) = X⊕Z you must imply that ⊕ forms a group operation this proof is split up into three parts

The symmetric difference is associative which means that
(X⊕Y)⊕(Y⊕Z)= (Y⊕Y)⊕(X⊕Z) --I think that is correct?--
or X⊕(Y⊕Z ) = (X⊕Y) ⊕Z

The symmetric difference of the same set yields an empty set, Y⊕Y= ∅

The symmetric difference of a set and empty set yields a Y⊕∅= Y

so (Y⊕Y) = ∅
∅ ⊕(X⊕Z)= X⊕Z
hence for any sets X,Y & Z ,(X⊕Y)⊕(Y⊕Z) = X⊕Z

THAT ALL CORRECT?
 
  • #5
22,129
3,298
ok here is my revised attempt at the answer with your help:

In order to prove that (X⊕Y)⊕(Y⊕Z) = X⊕Z you must imply that ⊕ forms a group operation this proof is split up into three parts

The symmetric difference is associative which means that
(X⊕Y)⊕(Y⊕Z)= (Y⊕Y)⊕(X⊕Z) --I think that is correct?--
or X⊕(Y⊕Z ) = (X⊕Y) ⊕Z

Well, you're using commutativity here (which is fine, but you must first show that you can do that). IF you don't want to do that, then

[tex](X\oplus Y)\oplus (Y\oplus Z)=X\oplus (Y\oplus Y)\oplus Z=X\oplus \emptyset \oplus Z=X\oplus Z[/tex]

is also fine...

The symmetric difference of the same set yields an empty set, Y⊕Y= ∅

The symmetric difference of a set and empty set yields a Y⊕∅= Y

so (Y⊕Y) = ∅
∅ ⊕(X⊕Z)= X⊕Z
hence for any sets X,Y & Z ,(X⊕Y)⊕(Y⊕Z) = X⊕Z

THAT ALL CORRECT?
 

Related Threads on Symmetric difference in sets

  • Last Post
Replies
19
Views
11K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
842
Replies
2
Views
746
Replies
3
Views
2K
Replies
7
Views
5K
Replies
6
Views
8K
Replies
27
Views
3K
Replies
12
Views
11K
  • Last Post
Replies
3
Views
1K
Top