# Tagged Questions

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### Cardinal Arithmetic and a permutation function.

I am working on the following problem and am having difficulties getting started: We define a permutation of $K$ to be any one-to-one function from $K$ onto $K$. We can then define the factorial ...
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### 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 ...
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### 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 $s_2$? ...
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### 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 ...
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### 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 ...
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### 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 $a∈A$, ...
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### 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 } ...