Symmetric Difference Proof for Sets A, B, and C

In summary, to prove that A%B=C iff B%C=A, we need to show that B%C is equal to B%(A%B). This can be done by taking x in (B%C)U(C/B) and showing that x is also in B%(A%B), which implies that B%C is a subset of B%(A%B). Then, we can use the given information that A%B=C to show that B%(A%B) is a subset of B%C, thus proving that B%C is equal to B%(A%B) and therefore equal to A.
  • #1
bedi
81
0

Homework Statement



Let % be the symmetric difference.
Prove that for any sets A, B, C;

A%B=C iff B%C=A iff A%C=B

Homework Equations



(I will use forwardslash as I cannot find the backslash on my keyboard.)

The Attempt at a Solution



Take x in A%B. Then x is either in A/B or in B/A and in C. Choose the former. Then x is in A and x is in C which implies x is in AnC. This is what I know.
Now, using these I will prove that B%C=A.
Take x in (B/C)U(C/B). Then x is either in the first or in the second. Actually it cannot be in the first because that would imply that x is not in C, which contradicts that x is in C. So x is in C/B. But I already know that x is in A, so nothing to prove in that direction. Take x in A... I don't know how to proceed.
 
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  • #2
hi bedi! :smile:
bedi said:
A%B=C iff B%C=A …

(to prove from left to right …)

you need to start "B%C = B%(A%B) = … " :wink:
 

Related to Symmetric Difference Proof for Sets A, B, and C

1. What is a symmetric difference proof?

A symmetric difference proof is a type of mathematical proof used to show that two sets are equal. It involves proving that the symmetric difference of the two sets is empty, meaning that there are no elements that exist in one set but not the other.

2. How is a symmetric difference proof different from other types of proofs?

A symmetric difference proof is different from other types of proofs because it specifically focuses on proving the equality of two sets, rather than proving a statement or equation to be true. It also uses the concept of the symmetric difference, which is not typically used in other types of proofs.

3. What is the process for conducting a symmetric difference proof?

The process for conducting a symmetric difference proof involves first defining the two sets that are being compared, then finding the symmetric difference of the two sets by finding the elements that are in one set but not the other. Next, the proof must show that this symmetric difference is empty by using logical steps and properties of sets.

4. Why is a symmetric difference proof important?

A symmetric difference proof is important because it provides a rigorous and logical way to show that two sets are equal. It is also a useful tool for solving problems in mathematics and other fields that involve sets and their relationships.

5. Are there any limitations to using a symmetric difference proof?

One limitation of using a symmetric difference proof is that it only works for proving the equality of two sets. It cannot be used to prove other types of statements or equations. Additionally, depending on the complexity of the sets being compared, a symmetric difference proof may be difficult to construct or may require a significant amount of steps.

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