Symmetric Difference if/then Proof

[pingi]

Hi there,

I'm trying to figure out proving the following:
if X oplus Y = Y oplus X then X = Y

In order to prove it, I need to use the symmetric difference associativity & other characteristics and identities.

Can you please give me a direction?
Please explain the answer as a teacher would, as my skills of proving this kind of arguments are poor.

Thanks! Pingi.

 

[mathman]

What is oplus? Also what kind of things are X and Y (sets, logical variables, other things)?

 

[pingi]

Hi mathman,

Sorry for being unclear about my question and thanks for directing me!

1. X and Y are sets.
2. 'oplus' is an add symbol in circle - ⊕, used to describe the symmetric difference of the two sets (with the XOR operation).

Pingi.

 

[alexfloo]

If it is referring to the symmetric difference or to the exclusive or operations, then I actually don't believe the statement you are trying to prove is true. Both of these operations are commutative, which means that:

X "oplus" Y = Y "oplus" X

regardless of what X and Y are.

If "oplus" means something different, however, then please explain it in more depth.

 

[blue_raver22]

Sure, I'd be happy to help you with this proof. First, let's define the symmetric difference operation, denoted by oplus, as the set of elements that are in either X or Y, but not both. In other words, if an element is in both X and Y, it is not part of the symmetric difference. So, X oplus Y can be written as (X ∪ Y) \ (X ∩ Y), where ∪ represents the union of sets and \ represents the set difference operation.

Now, let's assume that X oplus Y = Y oplus X. This means that the two sets have the same elements, since the symmetric difference is the same for both. In other words, (X ∪ Y) \ (X ∩ Y) = (Y ∪ X) \ (Y ∩ X).

Using the associative property of set union and intersection, we can rewrite this as (X \ Y) ∪ (Y \ X) = (Y \ X) ∪ (X \ Y). This shows that the elements in X that are not in Y are the same as the elements in Y that are not in X.

Now, let's assume that X ≠ Y. This means that there is at least one element in X that is not in Y and vice versa. So, (X \ Y) ∪ (Y \ X) ≠ ∅, where ∅ represents the empty set. However, since we have already established that (X \ Y) ∪ (Y \ X) = (Y \ X) ∪ (X \ Y), this would mean that (Y \ X) ∪ (X \ Y) ≠ ∅ as well. This is a contradiction, as the symmetric difference operation results in an empty set when the two sets have the same elements. Therefore, our assumption that X ≠ Y must be false, and thus X = Y.

In summary, we have shown that if X oplus Y = Y oplus X, then X = Y. This is because the elements in X that are not in Y are the same as the elements in Y that are not in X, and if X ≠ Y, there would be at least one element in the symmetric difference, which is not the case. Therefore, X must equal Y. I hope this explanation helps you understand the proof better. Keep practicing and you will improve your skills in proving arguments