Mixing
Prove that any set X has exactly one power set.
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Intuitively this makes sense, as the power set is the set of all subsets of X. So far I have assumed P,Q are two distinct power sets, then for any $a in P, a subset X$ so $a in Q$. Which gives $P subset Q$, the other direction gives $Q subset P$. But I am not sure it is complete?
elementary-set-theory
asked Jan 2 at 12:58
kappakappa
11
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Intuitively this makes sense, as the power set is the set of all subsets of X. So far I have assumed P,Q are two distinct power sets, then for any $a in P, a subset X$ so $a in Q$. Which gives $P subset Q$, the other direction gives $Q subset P$. But I am not sure it is complete?
elementary-set-theory
asked Jan 2 at 12:58
kappakappa
11
$endgroup$
add a comment |
$begingroup$
Intuitively this makes sense, as the power set is the set of all subsets of X. So far I have assumed P,Q are two distinct power sets, then for any $a in P, a subset X$ so $a in Q$. Which gives $P subset Q$, the other direction gives $Q subset P$. But I am not sure it is complete?
elementary-set-theory
asked Jan 2 at 12:58
kappakappa
11
$endgroup$
Intuitively this makes sense, as the power set is the set of all subsets of X. So far I have assumed P,Q are two distinct power sets, then for any $a in P, a subset X$ so $a in Q$. Which gives $P subset Q$, the other direction gives $Q subset P$. But I am not sure it is complete?
elementary-set-theory
elementary-set-theory
asked Jan 2 at 12:58
kappakappa
11
asked Jan 2 at 12:58
kappakappa
11
asked Jan 2 at 12:58
kappakappa
11
asked Jan 2 at 12:58
kappakappa
11
asked Jan 2 at 12:58
kappakappa
11
11
add a comment |
add a comment |
1 Answer
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The axiom of extensionality states sets are equal iff they have the same elements, which completes a proof there is at most one power set. That there is one is guaranteed by another axiom of most set theories, the power set axiom.
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
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1 Answer
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1 Answer
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$begingroup$
The axiom of extensionality states sets are equal iff they have the same elements, which completes a proof there is at most one power set. That there is one is guaranteed by another axiom of most set theories, the power set axiom.
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
$endgroup$
add a comment |
$begingroup$
The axiom of extensionality states sets are equal iff they have the same elements, which completes a proof there is at most one power set. That there is one is guaranteed by another axiom of most set theories, the power set axiom.
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
$endgroup$
add a comment |
$begingroup$
The axiom of extensionality states sets are equal iff they have the same elements, which completes a proof there is at most one power set. That there is one is guaranteed by another axiom of most set theories, the power set axiom.
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
$endgroup$
The axiom of extensionality states sets are equal iff they have the same elements, which completes a proof there is at most one power set. That there is one is guaranteed by another axiom of most set theories, the power set axiom.
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
answered Jan 2 at 13:03
J.G.J.G.
23.8k22538
23.8k22538
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