# The Probabilistic Method: Unbalancing Lights [closed]

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

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

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• Contains a link to a many hundred page pdf. Please type out the relevant details. – Graham Kemp Nov 29 '16 at 1:55

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 $y$.