Is the Symmetric Difference Problem Solved?

In summary, the conversation discusses two statements involving symmetric difference and the proof or contradiction of their validity. The speaker also mentions the definition of symmetric difference and a possible approach to proving or contradicting the statements. Finally, they ask for confirmation on the corresponding Venn diagrams.
  • #1
Yankel
395
0
Hello all,

For each of the following statements, I need to say if it is true or not, to prove if it is true or to contradict if not.

1)
\[A\bigtriangleup (B\cap C)=(A\bigtriangleup B)\cap (A\bigtriangleup C)\]

2)
\[A\cup (B\bigtriangleup C)=(A\cup B)\bigtriangleup (A\cup C)\]

Where
\[\bigtriangleup\]

is the symmetric difference.

I do know that:

\[A\bigtriangleup B=(A-B)\cup (B-A)\]

which is:

\[(A\cap B^{C})\cup (B\cap A^{C})\]How do I proceed from here?

Thank you !
 
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  • #2
Both equations can be refuted by taking $A$ to be the universal set.
 
  • #3
Is it correct that these are the corresponding Venn diagrams of both sides in the first statement?
(assuming your way, of making A universal).

View attachment 6444
 

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FAQ: Is the Symmetric Difference Problem Solved?

What is the Symmetric Difference Problem?

The Symmetric Difference Problem is a mathematical problem that involves finding the set of elements that are unique to two or more sets. It is often represented with the symbol (∆) and is also known as the "exclusive or" operation.

How is the Symmetric Difference Problem different from the Union and Intersection Problems?

The Union and Intersection Problems involve combining or finding common elements between sets, whereas the Symmetric Difference Problem focuses on finding the unique elements between sets. In other words, the Symmetric Difference Problem only includes elements that are not shared between sets.

What are some real-life applications of the Symmetric Difference Problem?

The Symmetric Difference Problem is commonly used in data analysis and programming. It can be used to identify differences between datasets, detect changes in data over time, and remove duplicate values from a dataset. It is also used in cryptography and error correction algorithms.

How is the Symmetric Difference Problem solved?

The Symmetric Difference Problem is typically solved by first finding the union of all the sets involved, and then subtracting the intersection of the sets. This leaves only the unique elements, which is the solution to the Symmetric Difference Problem.

Are there any limitations or challenges associated with the Symmetric Difference Problem?

One limitation of the Symmetric Difference Problem is that it only works with sets of data, and not with other data structures such as arrays or lists. Additionally, it can be challenging to solve for large datasets, as it requires a significant amount of computational power and time.

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