Mixing
Set theoretic proof involving union and intersection identity
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How to prove that $$Acap(Bcup C)=(Acap B)cup C implies Csubset A$$
without returning back to symbolic logic. I've tried expanding with the distributive identities but it's not very clear to me of how to proceed from that...
elementary-set-theory
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
$endgroup$
add a comment |
$begingroup$
How to prove that $$Acap(Bcup C)=(Acap B)cup C implies Csubset A$$
without returning back to symbolic logic. I've tried expanding with the distributive identities but it's not very clear to me of how to proceed from that...
elementary-set-theory
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
$endgroup$
$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
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@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35
add a comment |
$begingroup$
How to prove that $$Acap(Bcup C)=(Acap B)cup C implies Csubset A$$
without returning back to symbolic logic. I've tried expanding with the distributive identities but it's not very clear to me of how to proceed from that...
elementary-set-theory
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
$endgroup$
How to prove that $$Acap(Bcup C)=(Acap B)cup C implies Csubset A$$
without returning back to symbolic logic. I've tried expanding with the distributive identities but it's not very clear to me of how to proceed from that...
elementary-set-theory
elementary-set-theory
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
asked Jan 20 at 3:27
MacrophageMacrophage
1,181115
1,181115
$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
$begingroup$
@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35
add a comment |
$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
$begingroup$
@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35
$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
$begingroup$
@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35
$begingroup$
@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35
add a comment |
1 Answer
1
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oldest
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$begingroup$
Assume x in C. Then x in $(Acap B)cup C.$
Thus x in $Acap(Bcup C)$, x in A.
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Assume x in C. Then x in $(Acap B)cup C.$
Thus x in $Acap(Bcup C)$, x in A.
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
$endgroup$
add a comment |
$begingroup$
Assume x in C. Then x in $(Acap B)cup C.$
Thus x in $Acap(Bcup C)$, x in A.
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
$endgroup$
add a comment |
$begingroup$
Assume x in C. Then x in $(Acap B)cup C.$
Thus x in $Acap(Bcup C)$, x in A.
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
$endgroup$
Assume x in C. Then x in $(Acap B)cup C.$
Thus x in $Acap(Bcup C)$, x in A.
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
answered Jan 20 at 3:51
William ElliotWilliam Elliot
8,3372720
8,3372720
add a comment |
add a comment |
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$begingroup$
Try the contrapositive! Assume there is some element $x in C$ that is not in $A$, then show that it contradicts the equation on the LHS.
$endgroup$
– Zubin Mukerjee
Jan 20 at 3:33
$begingroup$
@ZubinMukerjee Thank you! That's a good point. I'll have to try it out.
$endgroup$
– Macrophage
Jan 20 at 3:35