To delve into this question, we first need to understand what a power set is. The power set of a set is the collection of all possible subsets of that set, including the empty set and the set itself. For instance, if we have a set A = {1, 2}, its power set, denoted as P(A), would be P(A) = {∅, {1}, {2}, {1, 2}}.
Now, let’s consider two sets, A and B, both possessing the same power set, P(A) = P(B). The main question we need to answer is whether this means that A = B.
The answer is yes, we can conclude that A and B are equal. This conclusion arises from the fact that a power set uniquely defines a set. If two sets have identical power sets, it implies that they contain the same elements.
To illustrate this, let’s analyze the implications of having the same power set:
- If A and B have identical power sets, every subset that can be formed from A must also be formable from B and vice versa.
- This reciprocal relationship suggests that any element in A must be present in B and any element in B must be present in A.
- Therefore, the sets A and B must consist of precisely the same elements, leading us to deduce that A = B.
In conclusion, yes, if two sets share the same power set, we can confidently assert that the two sets are indeed equal.