Is the empty set a strict order?
0
$begingroup$
I've read that it is a total order, but is it strict on the empty set? I think this may be vacuously true, but I don't know if that is true.
order-theory
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
$endgroup$
|
show 1 more comment
0
$begingroup$
I've read that it is a total order, but is it strict on the empty set? I think this may be vacuously true, but I don't know if that is true.
order-theory
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
$endgroup$
5
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
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I guess I don't really understand how to prove something is vacuously true.
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– Garmekain
Jun 4 '18 at 21:05
4
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Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
1
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
3
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11
|
show 1 more comment
0
0
0
0
$begingroup$
I've read that it is a total order, but is it strict on the empty set? I think this may be vacuously true, but I don't know if that is true.
order-theory
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
$endgroup$
I've read that it is a total order, but is it strict on the empty set? I think this may be vacuously true, but I don't know if that is true.
order-theory
order-theory
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
asked Jun 4 '18 at 21:01
GarmekainGarmekain
1,345720
1,345720
5
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
$begingroup$
I guess I don't really understand how to prove something is vacuously true.
$endgroup$
– Garmekain
Jun 4 '18 at 21:05
4
$begingroup$
Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
1
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
3
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11
|
show 1 more comment
5
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
$begingroup$
I guess I don't really understand how to prove something is vacuously true.
$endgroup$
– Garmekain
Jun 4 '18 at 21:05
4
$begingroup$
Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
1
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
3
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11
5
5
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
$begingroup$
I guess I don't really understand how to prove something is vacuously true.
$endgroup$
– Garmekain
Jun 4 '18 at 21:05
$begingroup$
I guess I don't really understand how to prove something is vacuously true.
$endgroup$
– Garmekain
Jun 4 '18 at 21:05
4
4
$begingroup$
Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
$begingroup$
Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
1
1
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
3
3
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11
|
show 1 more comment
1 Answer
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$begingroup$
From the comments above.
Yes, this is vacuously true.
In fact, any claim of the form "Every element of the empty set is [---]" is automatically true, while "some element of the empty set is [---]" is automatically false.
$endgroup$
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$begingroup$
From the comments above.
Yes, this is vacuously true.
In fact, any claim of the form "Every element of the empty set is [---]" is automatically true, while "some element of the empty set is [---]" is automatically false.
$endgroup$
add a comment |
1
$begingroup$
From the comments above.
Yes, this is vacuously true.
In fact, any claim of the form "Every element of the empty set is [---]" is automatically true, while "some element of the empty set is [---]" is automatically false.
$endgroup$
add a comment |
1
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$begingroup$
From the comments above.
Yes, this is vacuously true.
In fact, any claim of the form "Every element of the empty set is [---]" is automatically true, while "some element of the empty set is [---]" is automatically false.
$endgroup$
From the comments above.
Yes, this is vacuously true.
In fact, any claim of the form "Every element of the empty set is [---]" is automatically true, while "some element of the empty set is [---]" is automatically false.
answered Dec 4 '18 at 12:44
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Brahadeesh
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5
$begingroup$
Yes, it's vacuously true. Why do you doubt it?
$endgroup$
– saulspatz
Jun 4 '18 at 21:04
$begingroup$
I guess I don't really understand how to prove something is vacuously true.
$endgroup$
– Garmekain
Jun 4 '18 at 21:05
4
$begingroup$
Just check the handful of properties that define strict order. Essentially, any statement that starts with $forall cinemptyset$ is vacuously true
$endgroup$
– Hagen von Eitzen
Jun 4 '18 at 21:06
1
$begingroup$
A statement is vacuous if it doesn't apply to anything -- "vacuous" and "vacuum" have the same root. Such a statement must be true because there can't be any counterexample. So as Hagen von Eitzen has said, any statement about all the elements of the empty set must be vacuously true.
$endgroup$
– saulspatz
Jun 4 '18 at 21:10
3
$begingroup$
A minor amendation to saulspatz' comment: any universal claim about the elements of the emptyset. "Every element of the emptyset is [---]" is automatically true, while "some element of the emptyset is [---]" is automatically false.
$endgroup$
– Noah Schweber
Jun 4 '18 at 21:11