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Consider an example of flipping a coin infinitely many times. In class, the following notation was used:

$$\Omega=\{H,T\}^{\mathbb{N}}$$

Although I understand what $\Omega$ is supposed to look like, (infinite numerations of the infinite combinations of Heads and Tails), what is the sense/logic behind this notation?

Parcly Taxel
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Kwame Brown
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1 Answers1

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For sets $A$ and $B$, the notation $B^A$ is used for the set of functions from $A$ to $B$. (Possible reason for this is that if $A$ and $B$ are finite, then $\mid B^A \mid = \mid B \mid^{\mid A \mid}$, where $|S|$ is the number of elements in set $S$).

A sequence of coin tosses can be regarded as a function from $\mathbb{N}$ to $\{H,T\}$ where $1\mapsto$ first toss, etc.

paw88789
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  • Thanks a lot..However, why is the mapping from $\mathbb{N}$ to ${H,T}$? Using your notation, isn't set $B$ an infinite set? – Kwame Brown Sep 14 '16 at 15:57
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    If you have a sequence of heads and tails, say HHT..., then the mapping would be 1 to H; 2 to H; 3 to T; etc. Corresponding to the notation in my answer: ${H,T}$ is $B$ and $\mathbb{N}$ is $A$. – paw88789 Sep 14 '16 at 16:05
  • Merci beaucoup! – Kwame Brown Sep 14 '16 at 16:08