Show that if A is a subset of B, then the power set of A is a subset of the power set of B.
represents the empty set and the empty set does not contain any elemnets
The power set of S is the set of all subsets of S.
X is a subset of Y if every element of X is also an element of Y.
Notation: role="math" localid="1668425827447"
To proof :
If role="math" localid="1668425925367" contains only the empty set ,the proof is trivial as any power set contains the empty set .
If does not contain only the empty set , then there exists a set of the form in .
If the set containing only an element x is a set in the power set of , then the element x has to be an element of the set S.
If x is an element in a set S, then the set containing only that element x is a set in the power set of S.
We thus have derived that every element x in also has to be in .
By the definition of a subset, we know that is a subset of .
Hence the solution is,
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