Showing when a subset is an equality

  • Thread starter Mr Davis 97
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In summary, the question is asking under what conditions does equality hold for ##A \subseteq \mathcal{P} \bigcup A##. The attempt at a solution suggests that if A has n elements, then ##| \bigcup A | \ge n##, which would make ##| \mathcal{P} \bigcup A | \ge 2^n##. However, since ##\mathcal{P} \bigcup A## and ##\bigcup \mathcal{P} A## are different things, equality can never hold. The question then arises, what is ##\bigcup A##, the union of what if A is a single object?
  • #1
Mr Davis 97
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Homework Statement


In general, ##A \subseteq \mathcal{P} \bigcup A##. Under what conditions does equality hold?

Homework Equations

The Attempt at a Solution


I can't seem to figure this out. If ##A## has ##n## elements, then clearly ##| \bigcup A | \ge n##, which would mean that ##| \mathcal{P} \bigcup A | \ge 2^n##, right? In this case the cardinalities never seem to be the same, in which case equality can never hold.
 
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  • #2
## \mathcal{P} \bigcup A ## and ##\bigcup \mathcal{P} A ## are different things.
 
  • #3
mfb said:
## \mathcal{P} \bigcup A ## and ##\bigcup \mathcal{P} A ## are different things.
Oops. In both cases I meany ##\mathcal{P} \bigcup A##
 
  • #4
What is ##\bigcup A##? The union of what, if A is a single object?
 

FAQ: Showing when a subset is an equality

1. What is the definition of an equality in a subset?

An equality in a subset means that the elements in the subset are exactly the same as the elements in the original set. In other words, the subset contains all the elements of the original set, and no additional elements.

2. How can you prove that a subset is an equality?

To prove that a subset is an equality, you must show that all the elements in the subset are also present in the original set. This can be done by listing out all the elements in both sets and comparing them, or by using logical proofs such as contrapositive or contradiction.

3. Can a subset be an equality if it contains duplicate elements?

No, a subset cannot be an equality if it contains duplicate elements. An equality means that the subset must have the exact same elements as the original set, and duplicate elements would violate this condition.

4. Does the order of elements matter in determining an equality in a subset?

Yes, the order of elements does matter in determining an equality in a subset. Even if the same elements are present in both the subset and original set, if the order is different, they are not considered equal subsets.

5. Is an empty set considered an equality in a subset?

Yes, an empty set is considered an equality in a subset. This is because an empty set contains no elements, and therefore all the elements in the empty set are also present in the original set.

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