-5
votes
1answer
63 views

Dividing a set of n elements into k disjoint subsets.

I have been able to do the 1st part. I have not been able to prove the 2nd part. My attempt to the solution :- I took $k$ groups $ a_1, a_2, a_3…, a_k $ Let $a_1$ group has $b_1$ similarly so ...
1
vote
2answers
56 views

Is there always a bijection mapping one element of an infinite set onto another?

Let $S$ be an infinite set, and let $s_1$, $s_2$ be any two distinct elements of $S$. Then how to determine whether or not there is always a bijection of $S$ onto itself that maps $s_1$ onto $s_2$? ...
2
votes
1answer
265 views

Basic questions on permutation of sets (composition, inverse and signatures)

I am having trouble finding good resources to understand composition of permutation, I was wondering how can you go about multiplying (composition) of two permutations and how do they return the just ...
0
votes
1answer
120 views

Permutations & Functions

This is an assignment question I received a week ago. A function $f:\{1, 2, \dots ,n\} \to \{1, 2, \dots, n\}$ which is a bijection is also called a permutation. Let $P_n$ be the set of all ...
2
votes
2answers
215 views

Directed Graphs on Relations - Set Theory

These questions were from an assignment I had some time ago but the solutions were not provided. A permutation $P$ on a finite set $A$ is a binary relation with the property that for each $a∈A$, ...
0
votes
2answers
81 views

About $|\operatorname{Sym}(\Omega)|$ when $\Omega$ is an infinite set.

Here is a problem: Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$. I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } ...
0
votes
2answers
125 views

How can I figure out every permutation of sets of groups of four students in a class?

I'm trying to develop a grouping system that takes in a bunch of data for when particular students are available to meet and then spits out the "best" groups based on that data. Here I am defining ...
3
votes
2answers
366 views

Inversion Problem

I have a problem that says: Find the number of inversions in each of the following permutations of S = {1,2,3,4,5}: (a) 52134 (b) 45213 (c) 42135 In the text it doesn't do that great of a ...