Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?
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Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?
abstract-algebra permutations notation
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add a comment |
$begingroup$
Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?
abstract-algebra permutations notation
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8
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
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@Shaun Thanks I did it
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– BrianH
Dec 24 '18 at 20:48
add a comment |
$begingroup$
Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?
abstract-algebra permutations notation
$endgroup$
Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?
abstract-algebra permutations notation
abstract-algebra permutations notation
edited Dec 24 '18 at 20:19
Shaun
11.1k113688
11.1k113688
asked Dec 24 '18 at 20:13
BrianHBrianH
658
658
8
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48
add a comment |
8
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48
8
8
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48
$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}
text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.
$endgroup$
add a comment |
$begingroup$
The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).
$endgroup$
add a comment |
$begingroup$
$(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.
$endgroup$
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}
text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.
$endgroup$
add a comment |
$begingroup$
Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}
text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.
$endgroup$
add a comment |
$begingroup$
Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}
text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.
$endgroup$
Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}
text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.
answered Dec 24 '18 at 20:29
Jacky ChongJacky Chong
19.3k21129
19.3k21129
add a comment |
add a comment |
$begingroup$
The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).
$endgroup$
add a comment |
$begingroup$
The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).
$endgroup$
add a comment |
$begingroup$
The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).
$endgroup$
The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).
answered Dec 24 '18 at 20:23
ShaunShaun
11.1k113688
11.1k113688
add a comment |
add a comment |
$begingroup$
$(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.
$endgroup$
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
add a comment |
$begingroup$
$(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.
$endgroup$
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
add a comment |
$begingroup$
$(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.
$endgroup$
$(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.
answered Dec 24 '18 at 20:21
Maria MazurMaria Mazur
50.7k1362126
50.7k1362126
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
add a comment |
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
1
1
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
$endgroup$
– Eike Schulte
Dec 24 '18 at 22:22
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
$begingroup$
Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
$endgroup$
– Shaun
Dec 27 '18 at 13:36
add a comment |
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8
$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14
$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48