Proving a set statement
A, B, C are subsets of a set U:
A ⊆ B → A ∩ B $nsubseteq$ C (1)
A ⊆ B ∨ A ⊆ C (2)
A ∩ C ⊆ B (3)
I have to prove that this is valid:
A ∩ B $nsubseteq$ C (4)
It is recommended to use this in our proof:
X ⊆ Y ↔ X ∩ Y = X (5)
Should this be solved by mathematical induction or somehow else? I don't know what to do about that. Sorry for bad English. That is not my first language.
elementary-set-theory proof-explanation
asked Nov 28 '18 at 21:17
qwerty1
11
add a comment |
A, B, C are subsets of a set U:
A ⊆ B → A ∩ B $nsubseteq$ C (1)
A ⊆ B ∨ A ⊆ C (2)
A ∩ C ⊆ B (3)
I have to prove that this is valid:
A ∩ B $nsubseteq$ C (4)
It is recommended to use this in our proof:
X ⊆ Y ↔ X ∩ Y = X (5)
Should this be solved by mathematical induction or somehow else? I don't know what to do about that. Sorry for bad English. That is not my first language.
elementary-set-theory proof-explanation
asked Nov 28 '18 at 21:17
qwerty1
11
It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40
add a comment |
A, B, C are subsets of a set U:
A ⊆ B → A ∩ B $nsubseteq$ C (1)
A ⊆ B ∨ A ⊆ C (2)
A ∩ C ⊆ B (3)
I have to prove that this is valid:
A ∩ B $nsubseteq$ C (4)
It is recommended to use this in our proof:
X ⊆ Y ↔ X ∩ Y = X (5)
Should this be solved by mathematical induction or somehow else? I don't know what to do about that. Sorry for bad English. That is not my first language.
elementary-set-theory proof-explanation
asked Nov 28 '18 at 21:17
qwerty1
11
A, B, C are subsets of a set U:
A ⊆ B → A ∩ B $nsubseteq$ C (1)
A ⊆ B ∨ A ⊆ C (2)
A ∩ C ⊆ B (3)
I have to prove that this is valid:
A ∩ B $nsubseteq$ C (4)
It is recommended to use this in our proof:
X ⊆ Y ↔ X ∩ Y = X (5)
Should this be solved by mathematical induction or somehow else? I don't know what to do about that. Sorry for bad English. That is not my first language.
elementary-set-theory proof-explanation
elementary-set-theory proof-explanation
asked Nov 28 '18 at 21:17
qwerty1
11
asked Nov 28 '18 at 21:17
qwerty1
11
asked Nov 28 '18 at 21:17
qwerty1
11
asked Nov 28 '18 at 21:17
qwerty1
11
asked Nov 28 '18 at 21:17
qwerty1
11
11
It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40
add a comment |
It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40
It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40
It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40
add a comment |
1 Answer
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I'd solve it by cases on (2).
If $A subseteq B$, you're done by (1).
Suppose now that $A subseteq C$ (6).
By (6) we can say $A cap C = A$.
So by (3): $A cap C = A subseteq B$.
So $A subseteq B$ holds.
By (1): $A cap B nsubseteq C$.
answered Nov 28 '18 at 21:24
LuxGiammi
16410
add a comment |
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1 Answer
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1 Answer
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active
oldest
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I'd solve it by cases on (2).
If $A subseteq B$, you're done by (1).
Suppose now that $A subseteq C$ (6).
By (6) we can say $A cap C = A$.
So by (3): $A cap C = A subseteq B$.
So $A subseteq B$ holds.
By (1): $A cap B nsubseteq C$.
answered Nov 28 '18 at 21:24
LuxGiammi
16410
add a comment |
I'd solve it by cases on (2).
If $A subseteq B$, you're done by (1).
Suppose now that $A subseteq C$ (6).
By (6) we can say $A cap C = A$.
So by (3): $A cap C = A subseteq B$.
So $A subseteq B$ holds.
By (1): $A cap B nsubseteq C$.
answered Nov 28 '18 at 21:24
LuxGiammi
16410
add a comment |
I'd solve it by cases on (2).
If $A subseteq B$, you're done by (1).
Suppose now that $A subseteq C$ (6).
By (6) we can say $A cap C = A$.
So by (3): $A cap C = A subseteq B$.
So $A subseteq B$ holds.
By (1): $A cap B nsubseteq C$.
answered Nov 28 '18 at 21:24
LuxGiammi
16410
I'd solve it by cases on (2).
If $A subseteq B$, you're done by (1).
Suppose now that $A subseteq C$ (6).
By (6) we can say $A cap C = A$.
So by (3): $A cap C = A subseteq B$.
So $A subseteq B$ holds.
By (1): $A cap B nsubseteq C$.
answered Nov 28 '18 at 21:24
LuxGiammi
16410
answered Nov 28 '18 at 21:24
LuxGiammi
16410
answered Nov 28 '18 at 21:24
LuxGiammi
16410
answered Nov 28 '18 at 21:24
LuxGiammi
16410
16410
add a comment |
add a comment |
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It would be much better if you showed how much work you have done, and where exactly are you stuck. Which statements are premises, what do you want proven... so forth.
– Bertrand Wittgenstein's Ghost
Nov 29 '18 at 8:40