(A - B) union (B- A) = (A union B) - (A intersection B)

However, it is true for sets in general, as can be proven using set algebra.In summary, the statement (A - B) union (B- A) = (A union B) - (A intersection B) is true because the union of sets A and B contains all elements in A or B, and the sets (A - B) and (B - A) do not contain any elements from the other set. Therefore, their union is equal to the union of A and B. This can be proven using set algebra.
  • #1
yitriana
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Why is this true: (A - B) union (B- A) = (A union B) - (A intersection B)

wouldn't the union of A and B everything that is in A or B? And since A - B and B - A don't contain any elements from the other set, wouldn't the union of these be equal to union of A and B?

So wouldn't it make sense for it to be: (A - B) union (B- A) = (A union B)?

I don't think that A union B contains elements in A or B AND A and B.
 
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  • #3


yitriana said:
Why is this true: (A - B) union (B- A) = (A union B) - (A intersection B)
For an intuitive expalanation consider the Venn diagram here.

yitriana said:
So wouldn't it make sense for it to be: (A - B) union (B- A) = (A union B)?

This doesn't hold for A={1,2} and B={2,3}.
 

1. What is the meaning of "(A - B) union (B- A) = (A union B) - (A intersection B)"?

The expression "(A - B) union (B- A) = (A union B) - (A intersection B)" is a mathematical statement that represents the set of elements that are present in either A or B, but not both. It is known as the symmetric difference of sets A and B.

2. How is this equation useful in scientific research?

This equation is useful in scientific research as it allows us to compare and analyze data sets by identifying the unique elements present in each set. This can help in identifying patterns, similarities, and differences between data sets and can aid in making scientific conclusions.

3. Can you provide an example of how this equation can be applied in a scientific study?

Sure, let's say we have two sets of data: A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. The symmetric difference of these two sets would be (A - B) union (B- A) = {1, 2, 6, 7}. This tells us that the elements 1, 2, 6, and 7 are present in either A or B, but not both.

4. Are there any other ways to represent the symmetric difference of two sets?

Yes, the symmetric difference of sets A and B can also be represented as (A union B) - (A intersection B) or as (A - B) union (B - A). All three expressions are equivalent and represent the same set of elements.

5. Can this equation be extended to more than two sets?

Yes, the concept of symmetric difference can be extended to any number of sets. For example, the symmetric difference of three sets A, B, and C would be (A - B - C) union (B - A - C) union (C - A - B). This would represent the elements that are present in only one of the three sets.

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