Mixing
Clarification about a given axiom system.
$begingroup$
I am now currently studying Combinatorics of Finite Geometries. One problem asks if the given axiom system below is consistent or inconsistent.
- There are five points and six lines.
- Each point is in at most two lines.
- Each line contains two points.
Is the given axiom system consistent with a sample structure given below? 
My answer is no since axiom 2 will be violated. In particular there are points that are contained in six lines.
My questions are: (1) Am I correct? (2) If I am correct is there a possible structure that satisfies the given axiom system?
Thanks for the help.
geometry combinatorial-geometry
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
$endgroup$
add a comment |
$begingroup$
I am now currently studying Combinatorics of Finite Geometries. One problem asks if the given axiom system below is consistent or inconsistent.
- There are five points and six lines.
- Each point is in at most two lines.
- Each line contains two points.
Is the given axiom system consistent with a sample structure given below?
My answer is no since axiom 2 will be violated. In particular there are points that are contained in six lines.
My questions are: (1) Am I correct? (2) If I am correct is there a possible structure that satisfies the given axiom system?
Thanks for the help.
geometry combinatorial-geometry
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
$endgroup$
add a comment |
$begingroup$
I am now currently studying Combinatorics of Finite Geometries. One problem asks if the given axiom system below is consistent or inconsistent.
- There are five points and six lines.
- Each point is in at most two lines.
- Each line contains two points.
Is the given axiom system consistent with a sample structure given below?
My answer is no since axiom 2 will be violated. In particular there are points that are contained in six lines.
My questions are: (1) Am I correct? (2) If I am correct is there a possible structure that satisfies the given axiom system?
Thanks for the help.
geometry combinatorial-geometry
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
$endgroup$
I am now currently studying Combinatorics of Finite Geometries. One problem asks if the given axiom system below is consistent or inconsistent.
- There are five points and six lines.
- Each point is in at most two lines.
- Each line contains two points.
Is the given axiom system consistent with a sample structure given below?
My answer is no since axiom 2 will be violated. In particular there are points that are contained in six lines.
My questions are: (1) Am I correct? (2) If I am correct is there a possible structure that satisfies the given axiom system?
Thanks for the help.
geometry combinatorial-geometry
geometry combinatorial-geometry
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
edited Jan 12 at 4:59
Jr Antalan
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
asked Jan 9 at 10:52
Jr AntalanJr Antalan
1,2881822
1,2881822
add a comment |
add a comment |
1 Answer
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$begingroup$
(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.
(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.
answered Jan 9 at 11:04
bofbof
51.4k557120
$endgroup$
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
add a comment |
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1 Answer
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$begingroup$
(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.
(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.
answered Jan 9 at 11:04
bofbof
51.4k557120
$endgroup$
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
add a comment |
$begingroup$
(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.
(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.
answered Jan 9 at 11:04
bofbof
51.4k557120
$endgroup$
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
add a comment |
$begingroup$
(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.
(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.
answered Jan 9 at 11:04
bofbof
51.4k557120
$endgroup$
(1) You are correct. I hope your teacher isn't the one who thought the system in the picture satisfies the axioms.
(2) No, the given axiom system is not satisfiable. How many pairs $(p,L)$ are there, consisting of a point $p$ which is on a line $L$? By axioms 1 and 2, there are at most $10$ such pairs; but by axioms 1 and 3, there must be $12$ of them. The axioms are inconsistent.
answered Jan 9 at 11:04
bofbof
51.4k557120
answered Jan 9 at 11:04
bofbof
51.4k557120
answered Jan 9 at 11:04
bofbof
51.4k557120
answered Jan 9 at 11:04
bofbof
51.4k557120
51.4k557120
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
add a comment |
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
$begingroup$
Thanks @bof. All clear to me now.
$endgroup$
– Jr Antalan
Jan 9 at 11:40
add a comment |
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