Proof of a set union and intersection

In summary: Keep up the good work!In summary, the conversation involved the speaker struggling with an assignment involving union and intersection of sets. They were unsure of how to interpret the definition of intersection but were able to successfully solve the problem with the help of their professor's explanation. The conversation ended on a positive note with the speaker feeling confident in their understanding of the topic.
  • #1
2
0
Hello!
Lately, I've been struggling with this assignment. (angle brackets represent closed interval)
Screenshot_20211116_184448(1).png


I figured out that:

a)
union = R
intersection = {0}
b)
union = (0, 2)
intersection = {1}

I asked my prof about this and she explained to me that it should be shown that if a set is an intersection of sets, then it belongs to each of those sets and, conversely, nothing else belongs to the intersection, so every other element does not belong to at least one of those sets. But I don't really know how to interpret this or where to even start. (normally, when proving the equality of two sets, I would try to prove that A⊆B and B⊆A, but I don't see how that's applicable here).

Thank you for your help!
 
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  • #2
bargaj said:
angle brackets represent closed interval
Never seen this notation. Usually closed intervals are denoted by square brackets.

bargaj said:
a)
union = R
intersection = {0}
b)
union = (0, 2)
intersection = {1}
I agree.

bargaj said:
I asked my prof about this and she explained to me that it should be shown that if a set is an intersection of sets, then it belongs to each of those sets and, conversely, nothing else belongs to the intersection, so every other element does not belong to at least one of those sets.
So we reached a consensus, right?

bargaj said:
But I don't really know how to interpret this or where to even start.
You not only started, but also finished solving these problems.The professor just described the definition of intersection, which you have successfully applied, so you must have understood it.
 

1. What is the proof of a set union?

The proof of a set union is a mathematical demonstration that shows that the union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in either A or B (or both).

2. How is the proof of a set union different from the proof of a set intersection?

The proof of a set intersection is a mathematical demonstration that shows that the intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are in both A and B. This is different from the proof of a set union, which shows the elements that are in either A or B.

3. What are the key steps in proving a set union?

The key steps in proving a set union include defining the sets A and B, showing that the union of A and B contains all elements in A and B, and proving that there are no additional elements in the union.

4. How can you prove that two sets are equal using the proof of set union and intersection?

If two sets A and B have the same elements, then their union and intersection will also be equal. This can be proven by showing that the union and intersection of A and B contain the same elements and that there are no additional elements in either set.

5. Can the proof of set union and intersection be applied to more than two sets?

Yes, the proof of set union and intersection can be extended to any number of sets. The union of multiple sets will contain all elements that are in any of the sets, while the intersection will contain only the elements that are in all of the sets.

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