Intuition behind multiplying (or composing) permutations. I'm trying to grasp the intuition for per

letumsnemesislh 2022-07-08 Answered
Intuition behind multiplying (or composing) permutations.
I'm trying to grasp the intuition for permutations and their multiplication. So far this has been my intuitive understanding: A permutation is merely a shuffling of the symbols. Take for example σ , π S 4 given by, σ = ( 1 2 3 4 3 2 1 4 ) and π = ( 1 2 3 4 2 4 1 3 )
I could rewrite them as a 4-tuple: σ = ( 3 , 2 , 1 , 4 ) and π = ( 2 , 4 , 1 , 3 ) as permutations of {1,2,3,4} and so (#) π σ = ( 2 , 4 , 1 , 3 ) ( 3 , 2 , 1 , 4 ) = ( 1 , 4 , 2 , 3 )
I understand how to get the result. I know how to multiply (or compose) two permutations.
My Question: What happened in equation # and what's going on intuitively? What shuffled around when composition happened? What does the result of product mean with respect to π and σ?
You can still ask an expert for help

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Elias Flores
Answered 2022-07-09 Author has 24 answers
Step 1
We can formulate the composition rule of permutations given in the form
π σ = ( 2 , 4 , 1 , 3 ) ( 3 , 2 , 1 , 4 ) = ( 1 , 4 , 2 , 3 )
as
- Select the items of the left-hand permutation π in the order given by the items of the right-hand permutation σ.
Step 2
In the current example we have the order of the items of the right-hand permutation σ as
σ = ( 3 , 2 , 1 , 4 ) third, second, first and fourth item of  π resulting in π σ = ( 1 , 4 , 2 , 3 )

We have step-by-step solutions for your answer!

Raul Walker
Answered 2022-07-10 Author has 7 answers
Step 1
I'd say that it happens what the definition says it should happen: if you say that f = ( f 1 , f 2 , f 3 , f 4 ) is the function such that f ( i ) = f i , and if you say that π σ is the function such that ( π σ ) ( i ) = π ( σ ( i ) ), then the representation of π σ will be given by ( π σ ) i = π σ i .
Step 2
Of course, you should be careful whe you use this the one-line notation because you are using essentially the same notation as the very common cycle notation.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-08-02
Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols:
asked 2021-07-28

Let A, B, and C be sets. Show that (AB)C=(AC)(BC)
image

asked 2021-08-18
Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b)R if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
asked 2020-11-09
Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.
asked 2022-06-25
Battleship placement proving that the number of battleships is divisible by 3
We have a grid of 6 columns and x number of rows. All battleships are three units long and can be placed like this: 1 or like this: 2
Where the entire grid is filled with ships with no square units left unfilled, I'm interested in showing that the number of ships positioned as seen in (2) must be divisible by 3; the number of ships placed upright is divisible by 3.
I noticed that the restriction on the number of columns to be 6 means that each column can only have up to two of the ships of type (1), I can show that the number of square units remaining is divisible by 3 but I know that this does not imply the actual number of ships is divisible by 3.
asked 2022-05-22
A vertex of a minimum vertex cut has a neighbor in every component
I'm trying to understand the solution for the following problem: Prove that κ ( G ) = κ ( G ) when G is a simple graph with Δ ( G ) 3.
The solution goes like this: Let S be the minimum vertex cut, | S | = κ ( G ). Since κ ( G ) κ ( G ) always, we need only provide an edge cut of size |S|. Let H 1 and H 2 be two components of G-S. Since S is a minimum vertex cut, each v S has a neighbor in H 1 and a neighbor in H 2 . The solution conntinues from here...I really have no clue why the part "Since S is a minimum vertex cut, each v S has a neighbor in H 1 and a neighbor in H 2 ." is true.
I tried to show it by contradiction but haven't gotten very far: Suppose otherwise; then there exists a vertex v S which doesn't have a neighbor in H 1 . We can assume that G is connected, hence there is a path joining v with u V ( H 1 )... and I'm stuck. How do I proceed from here? Or should I try to prove this in a completely different way?
asked 2022-09-07
Are these two statements equivalent?
Express the statement that no one has more than three grandmothers.
G(x, y) : x is the grandmother of y
y ( ( a b c d , ( G ( a , y ) G ( b , y ) G ( c , y ) G ( d , y ) ) ) ( a = b   a = b   a = c   a = d   b = c   b = d   c = d ) )
This is my solution. What I am trying to say is that if there exists a person y (anyone) who has four grandmothers then at least two of those grandmothers must be the same.
Is this correct?
The books solution is this:
y ( ¬ a b c d , ( a b   a b   a c   a d   b c   b d   c d ( G ( a , y ) G ( b , y ) G ( c , y ) G ( d , y ) )
I am thinking this means: For all persons y, there does not exist four different people each of whom is the grandmother of y.
It seems as if mine is simple the negation of his statement, where
¬ p q = ¬ q p

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question