Mex (mathematics)

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In combinatorial game theory, the mex, or "minimum excludant", of a set of ordinals denotes the smallest ordinal not contained in the set.

Some examples:

\mbox{mex}(\left \{1, 2, 3\right \}) = 0
\mbox{mex}(\left \{0, 1, 4, 7, 12\right \}) = 2
\mbox{mex}(\left \{0, 1, 2, 3, \ldots\right \}) = \omega
\mbox{mex}(\left \{0, 2, 4, 6, \ldots\right \}) = 1
\mbox{mex}(\left \{0, 1, 2, 3, \ldots, \omega\right \}) = \omega+1

where ω is the limit ordinal for the natural numbers.

In the Sprague-Grundy theory the minimum excluded ordinal plays a dominant role in determining the Nimbers of combinatorial games.


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