Prove that for $A, B, C in mathcal{P}(mathbb{N})$ if sets $A triangle B$ and $B triangle C$ are finite then...
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Like in the title, we need to prove that:
$$forall A, B,C in mathcal{P}(mathbb{N}) quad (A triangle B text{ is finite } wedge B triangle C text{ is finite}) implies A triangle C text{ is finite } $$
We can write the definition: $A triangle B = A setminus B cup B setminus A $. Now we can see that the sets $A setminus B, C setminus B, B setminus A, B setminus C$ are finite. We have to prove that also $A setminus C, C setminus A$ is finite.
I tried to write $C = (C setminus B) cup (B cap C)$, $A = (A setminus B) cup (B cap A)$ so we could prove $(B cap C) triangle (B cap A)$ is finite, but it doesn't bring me anywhere.
elementary-set-theory
asked Dec 29 '18 at 15:17
VereXVereX
996
$endgroup$
add a comment |
$begingroup$
Like in the title, we need to prove that:
$$forall A, B,C in mathcal{P}(mathbb{N}) quad (A triangle B text{ is finite } wedge B triangle C text{ is finite}) implies A triangle C text{ is finite } $$
We can write the definition: $A triangle B = A setminus B cup B setminus A $. Now we can see that the sets $A setminus B, C setminus B, B setminus A, B setminus C$ are finite. We have to prove that also $A setminus C, C setminus A$ is finite.
I tried to write $C = (C setminus B) cup (B cap C)$, $A = (A setminus B) cup (B cap A)$ so we could prove $(B cap C) triangle (B cap A)$ is finite, but it doesn't bring me anywhere.
elementary-set-theory
asked Dec 29 '18 at 15:17
VereXVereX
996
$endgroup$
1
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27
add a comment |
$begingroup$
Like in the title, we need to prove that:
$$forall A, B,C in mathcal{P}(mathbb{N}) quad (A triangle B text{ is finite } wedge B triangle C text{ is finite}) implies A triangle C text{ is finite } $$
We can write the definition: $A triangle B = A setminus B cup B setminus A $. Now we can see that the sets $A setminus B, C setminus B, B setminus A, B setminus C$ are finite. We have to prove that also $A setminus C, C setminus A$ is finite.
I tried to write $C = (C setminus B) cup (B cap C)$, $A = (A setminus B) cup (B cap A)$ so we could prove $(B cap C) triangle (B cap A)$ is finite, but it doesn't bring me anywhere.
elementary-set-theory
asked Dec 29 '18 at 15:17
VereXVereX
996
$endgroup$
Like in the title, we need to prove that:
$$forall A, B,C in mathcal{P}(mathbb{N}) quad (A triangle B text{ is finite } wedge B triangle C text{ is finite}) implies A triangle C text{ is finite } $$
We can write the definition: $A triangle B = A setminus B cup B setminus A $. Now we can see that the sets $A setminus B, C setminus B, B setminus A, B setminus C$ are finite. We have to prove that also $A setminus C, C setminus A$ is finite.
I tried to write $C = (C setminus B) cup (B cap C)$, $A = (A setminus B) cup (B cap A)$ so we could prove $(B cap C) triangle (B cap A)$ is finite, but it doesn't bring me anywhere.
elementary-set-theory
elementary-set-theory
asked Dec 29 '18 at 15:17
VereXVereX
996
asked Dec 29 '18 at 15:17
VereXVereX
996
asked Dec 29 '18 at 15:17
VereXVereX
996
asked Dec 29 '18 at 15:17
VereXVereX
996
asked Dec 29 '18 at 15:17
VereXVereX
996
996
1
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27
add a comment |
1
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27
1
1
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27
add a comment |
1 Answer
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$begingroup$
Hint: try and guess what's
$$
(Amathbin{triangle}B)mathbin{triangle}(Bmathbin{triangle}C)
$$
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
$endgroup$
add a comment |
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$begingroup$
Hint: try and guess what's
$$
(Amathbin{triangle}B)mathbin{triangle}(Bmathbin{triangle}C)
$$
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
$endgroup$
add a comment |
$begingroup$
Hint: try and guess what's
$$
(Amathbin{triangle}B)mathbin{triangle}(Bmathbin{triangle}C)
$$
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
$endgroup$
add a comment |
$begingroup$
Hint: try and guess what's
$$
(Amathbin{triangle}B)mathbin{triangle}(Bmathbin{triangle}C)
$$
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
$endgroup$
Hint: try and guess what's
$$
(Amathbin{triangle}B)mathbin{triangle}(Bmathbin{triangle}C)
$$
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
answered Dec 29 '18 at 15:45
egregegreg
184k1486206
184k1486206
add a comment |
add a comment |
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1
$begingroup$
Prove first that $ADelta Csubseteq (ADelta B)cup(BDelta C)$.
$endgroup$
– Andreas Blass
Dec 29 '18 at 15:27