Symmetric difference in sets

In summary, the proof of (X⊕Y)⊕(Y⊕Z) = X⊕Z involves showing that the symmetric difference is associative and that the symmetric difference of a set and an empty set yields the original set. This implies that the symmetric difference forms a group operation and thus (X⊕Y)⊕(Y⊕Z) = X⊕Z is true for any sets X, Y, and Z.
  • #1
Uiiop
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 I am 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 don't have a clue how to show that properly in subsets etc that why i need some help
 
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  • #3
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
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
Uiiop said:
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 to Symmetric difference in sets

1. What is symmetric difference in sets?

Symmetric difference in sets refers to the elements that are present in one set but not the other. It is essentially the union of two sets minus the intersection of the two sets.

2. How is symmetric difference represented?

Symmetric difference can be represented using the symbol Δ or by using the keyword "symmetric_difference" in programming languages such as Python.

3. What is the relationship between symmetric difference and symmetric complement?

Symmetric difference and symmetric complement are closely related concepts. While symmetric difference is the set of elements that are in one set but not the other, symmetric complement is the set of elements that are not in either set. In other words, symmetric complement is the set of elements that are present in both sets.

4. How is symmetric difference different from set difference?

The main difference between symmetric difference and set difference is that symmetric difference includes elements that are present in either set, while set difference only includes elements that are present in one set but not the other.

5. Can the symmetric difference of two sets be an empty set?

Yes, the symmetric difference of two sets can be an empty set if the two sets have the same elements or if they have no common elements.

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