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I am reading The Probabilisic Method by Noga Alon and Joel Spencer (PDF available here: http://nguyen.hong.hai.free.fr/EBOOKS/SCIENCE%20AND%20ENGINEERING/MATHEMATIQUE/PROBABILITY/The_Probabilistic_Method.pdf). I am stuck at the example "Unbalancing lights" on page 212. In the fourth equation on this page, I don't unterstand the First equality: way can we rewrite the max over x and y as the max of a max? It makes sense to read the Wohle Paragraph. Thereare, I haven't included the equation here directly, singe it would make Little sense

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closed as off-topic by Shailesh, Graham Kemp, user137731, suomynonA, Leucippus Nov 29 '16 at 4:09

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  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shailesh, Community, suomynonA, Leucippus
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  • $\begingroup$ Contains a link to a many hundred page pdf. Please type out the relevant details. $\endgroup$ – Graham Kemp Nov 29 '16 at 1:55
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In general, $$\max_{x,y} f(x,y) = \max_x \max_y f(x,y) = \max_y \max_x f(x,y).$$

  • $\max_{x,y}f(x,y)$ is the maximum over all $x$ and $y$.
  • $\max_x \max_y f(x,y)$ can be interpreted as first finding the maximum value of the one-dimensional function $f(x,\cdot)$ for each $x$, and then taking a maximum over $x$.
  • $\max_y \max_x f(x,y)$ can be interpreted as first finding the maximum value of the one-dimensional function $f(\cdot,y)$ for each $y$, and then taking a maximum over $y$.
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