- #1
gutnedawg
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Can someone explain to me how to show (x\y) union (y\x) = (x union y) \ (y union x) using only the main set theory laws for union, intersections and difference.
Symmetric Difference Explanation
- Thread starter gutnedawg
- Start date
In summary, the professor suggests using the absorption laws to show that x\y is true. You can use Venn diagrams and translation to logic to better understand set operations.
- #2
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I have a feeling that this is a homework problem, so I'm not going to tell you a lot.
How do you usually show the equality of sets?
How do you usually show the equality of sets?
- #3
tiny-tim
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hi gutnedawg! 
shouldn't that be (x union y) \ (x intersection y) ?
anyway, start by writing x\y in terms of unions or intersections …
shouldn't that be (x union y) \ (x intersection y) ?
anyway, start by writing x\y in terms of unions or intersections …
what do you get?
- #4
gutnedawg
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yea sorry I meant to write union and then intersection not union and uniontiny-tim said:hi gutnedawg!
shouldn't that be (x union y) \ (x intersection y) ?
anyway, start by writing x\y in terms of unions or intersections …
what do you get?
this is not a homework problem this problem was posted in lecture for us to try on our own and I'm having trouble starting it
EDIT how could I write x\y in terms of intersections and unions... The professor suggested using the absorption laws but I'm not sure how to go on from rewriting x and y with the absorption laws
- #5
verty
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You've just got to apply the laws in different orders until you find the order that works. Trial and error, I'm afraid.
- #6
gutnedawg
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vertigo said:
care to give any hints?
- #7
tiny-tim
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hi gutnedawg!
(just got up :zzz: …)
(what are the absorption laws?)
(just got up :zzz: …)
gutnedawg said:
(what are the absorption laws?
)x\y = x intersection not-y
- #8
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General advice,
Set operations exactly mirror first order logic. Translate any set to the statement that some antecedent object is in the set. X --> "p is in X".
Then you can use your verbal skills to parse set operations...
x\y -> " p is in x and not in y".
A Venn diagram is also quite useful. Between Venn diagrams and translation to logic you can utilize your visual and verbal skills to better understand the set operations.
Finally ask yourself why your Professor didn't just tell you the answer and then trust his judgment. (Hint, you can't learn to swing a golf club by just watching the Pro's.)
Set operations exactly mirror first order logic. Translate any set to the statement that some antecedent object is in the set. X --> "p is in X".
Then you can use your verbal skills to parse set operations...
x\y -> " p is in x and not in y".
A Venn diagram is also quite useful. Between Venn diagrams and translation to logic you can utilize your visual and verbal skills to better understand the set operations.
Finally ask yourself why your Professor didn't just tell you the answer and then trust his judgment. (Hint, you can't learn to swing a golf club by just watching the Pro's.)
Related to Symmetric Difference Explanation
1. What is a symmetric difference in mathematics?
A symmetric difference is a mathematical operation that compares sets of elements and returns the elements that are unique to each set. In other words, it returns the elements that are present in one set but not the other.
2. How is symmetric difference represented in mathematical notation?
The symmetric difference is usually represented by the symbol ∆ or ⊕. For example, if we have two sets A and B, the symmetric difference can be written as A ∆ B or A ⊕ B.
3. What is the difference between symmetric difference and set difference?
The symmetric difference returns the elements that are unique to each set, while the set difference returns the elements that are only present in one set and not the other. In other words, the symmetric difference includes elements from both sets, while the set difference only includes elements from one set.
4. How is symmetric difference used in real-world applications?
Symmetric difference can be useful in data analysis and statistics, where it can help identify differences between two groups or datasets. It can also be used in cryptography and coding theory to compare and manipulate binary data.
5. What is an example of using symmetric difference in everyday life?
An example of using symmetric difference in everyday life is when comparing the preferences of two groups of people. For instance, if we have a group of people who prefer chocolate and a group who prefer vanilla, the symmetric difference would return the people who have a preference for only one flavor, while the set difference would return those who have a preference for either chocolate or vanilla, but not both.
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