咨询：为何{0}不是{0,1}幂集的子集？

Hey there, let's unpack this confusion step by step—this is a super common mix-up when first learning about power sets and subsets, so you're not alone!

First, let's restate the key definitions clearly to set the stage:

• The power set of a set ( S ), written ( \mathcal{P}(S) ), is the set of all subsets of ( S ). For ( S = {0,1} ), its subsets are:
  • The empty set: {}
  • The singleton set containing only 0: {0}
  • The singleton set containing only 1: {1}
  • The full set itself: {0,1}
    So ( \mathcal{P}({0,1}) = {{}, {0}, {1}, {0,1}} )—it looks like you missed {1} and {0,1} as elements of the power set earlier, which was part of the confusion!

Now, the critical rule to remember: A set ( A ) is a subset of set ( B ) if every element of ( A ) is also an element of ( B ). Let's apply this to each of the candidates:

• {0}: The only element of this set is the number 0. Now look at ( \mathcal{P}({0,1}) )—its elements are sets, not raw numbers. 0 is not an element of ( \mathcal{P}({0,1}) ), so {0} cannot be a subset of the power set.
• {{0}}: The only element of this set is {0}, which is an element of ( \mathcal{P}({0,1}) ). Since all elements of {{0}} are in the power set, it's a valid subset.
• {}: By definition, the empty set is a subset of every set—there are no elements to check, so this automatically counts.
• {{}}: The only element here is {}, which is an element of ( \mathcal{P}({0,1}) ). So this is also a valid subset.

Your confusion likely came from mixing up "elements of the power set" and "subsets of the original set". The power set doesn't contain numbers like 0 or 1—it contains sets that are subsets of {0,1}. So when checking if something is a subset of the power set, you're verifying whether its elements are those subset-sets, not the original numbers from {0,1}.

内容的提问来源于stack exchange，提问作者JobHunter69