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Let S be the set of all 52 cards. For our purposes here, we consider a "deck" as a permutation of these 52 cards, $(c_1, c_2,...,c_{52})$ where $c_1$ is the first card, $c_2$ the second card and so on. We let $Omega$ be the set of all possible decks.

a) Is $Omega$ a subset of $S^{52}$ and is $S^{52}$ a subset of $Omega%$

I think they are equal and therefore subsets of each other, but is $S^{52}$ standard notation?

b) A deck of cards is distributed to four players - A,B,C and D, each receiving $13$. The set of cards each player receives is called their hand. Let $mathcal{A}$ be the hand of player A. Is $mathcal{A}$ a subset of $S^{52}, S^{13} text{ or }Omega$?

I am again unsure of notation with the S's, but is it a subset of $Omega$ or would each subset of $Omega$ need to have 52 elements?

c) The cards are distributed in the cycle A-B-C-D-A-B-C-D.... Given a deck, $(c_1,...,c_{52})$ what is $mathcal{A}$?

Clearly, $(c_1,c_5,c_9,c_{13},...,c_{49})$

d) What is the subset of $Omega$ corresponding to player A has the ace of diamonds?

Would this just be $(c_1, c_2,...,c_{52})$ where the ace of diamonds is $c_i$ and $i equiv 1 text{ mod }4$

Any help or clarification of notation will be much appreciated. Thanks.

a)

If $X$ and $Y$ are sets then $Y^X$ denotes the set of functions $Xto Y$. Secondly $52$ can be interpreted as the set ${0,1,2,3,dots,51}$ so that $S^{52}$ can be looked at as the set of functions ${0,1,2,3,dots,51}to S$.

It does harm to replace $52={0,1,2,3,dots,51}$ by $[52]={1,2,3,dots,51,52}$ but in that case I would prefer the notation $S^{[52]}$.

Every deck can be recognized as an element of $S^{[52]}$ but not vice versa. Actually the decks can be identified as the bijections. So if $Omega$ denotes the set of all possible decks then $Omega$ is a proper subset of $S^{[52]}$ defined by: $$Omega={omegain S^{[52]}mid omegatext{ is bijective}}subsetneq S^{[52]}$$
Let me remark here that you can still take $S^{[52]}$ as the "set of outcomes" but in that case you must be aware of the fact that there are outcomes with $P({omega})=0$.

Any element $omegainOmega$ can be denoted as $(omega(1),omega(2),dots, omega(52))$.

b) and c)

You can define:

• $A={4k-3mid kin{1,2,dots,13}}$


• $B={4k-2mid kin{1,2,dots,13}}$


• $C={4k-1mid kin{1,2,dots,13}}$


• $D={4kmid kin{1,2,dots,13}}$

These sets are disjoint, covering and equinumberable subsets of $[52]$.

Then every $omegainOmega$ induces "hands" $mathcal A(omega),mathcal B(omega),mathcal C(omega),mathcal D(omega)$.

Here e.g. $mathcal A(omega)=omegaupharpoonleft A$ and can be denoted as $(omega(1),omega(5),dots,omega(49))$.

So actually $mathcal A$ is the function $OmegatoOmega_A={omegaupharpoonleft Amid omegainOmega}$ prescribed by $omegamapstoomegaupharpoonleft A$.

d)

Let it be that $c_k$ is the ace of diamonds.

Then ${text{A receives ace of diamonds}}={omegainOmegamid c_kintext{image of }mathcal A(omega)}$.