<div class="floatright"><a href="https://en.wikiversity.org/wiki/User:Watchduck/hat#Reverse_binary" title="The subsets of V_4 are numbered in reverse binary order."><img alt="The subsets of V_4 are numbered in reverse binary order." src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Little-endian_binary.svg/100px-Little-endian_binary.svg.png" decoding="async" width="100" height="179" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Little-endian_binary.svg/150px-Little-endian_binary.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Little-endian_binary.svg/200px-Little-endian_binary.svg.png 2x" data-file-width="819" data-file-height="1467"></a></div>
<p>The elements of <a href="https://en.wikipedia.org/wiki/en:Hereditarily_finite_set" class="extiw" title="w:en:Hereditarily finite set">V<sub>4</sub></a> - respectively the subsets of V<sub>3</sub> - ordered in a <a href="https://en.wikipedia.org/wiki/en:Hasse_diagram" class="extiw" title="w:en:Hasse diagram">Hasse diagram</a> by <a href="https://en.wikipedia.org/wiki/en:subset" class="extiw" title="w:en:subset">inclusion</a>.
</p><p>V<sub>4</sub> =  <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>^4(<a href="https://en.wikipedia.org/wiki/en:empty_set" class="extiw" title="w:en:empty set">{}</a>) = <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>( <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>( <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>(<a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>(<a href="https://en.wikipedia.org/wiki/en:empty_set" class="extiw" title="w:en:empty set">{}</a>)))) is the <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">powerset</a> of the <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">power set</a> of the <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">power set</a> of the <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">power set</a> of the <a href="https://en.wikipedia.org/wiki/en:empty_set" class="extiw" title="w:en:empty set">empty set</a>.
</p><p>As many brackets are usually hard to understand, the sets are defined by circles instead of brackets. The circle containing nothing stands for the empty set.
</p><p>The universe <small>(the thing all shown sets are a <a href="https://en.wikipedia.org/wiki/em:subset" class="extiw" title="w:em:subset">subset</a> from)</small> in this Hasse diagram is <a href="https://en.wikipedia.org/wiki/en:power_set" class="extiw" title="w:en:power set">P</a>^3(<a href="https://en.wikipedia.org/wiki/en:empty_set" class="extiw" title="w:en:empty set">{}</a>) and expressed in set brackets it looks like this: 
</p><p>{ {} , {{}} , {{{}}} , {{},{{}}} }
</p>
This system of sets can be used for a set theoretic definition of <a href="https://en.wikipedia.org/wiki/en:Boolean_function" class="extiw" title="w:en:Boolean function">Boolean functions</a>.
<table><tbody><tr><td style="padding: 15px;"><a href="//commons.wikimedia.org/wiki/File:Watchduck.svg" class="image"><img alt="Watchduck.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/40px-Watchduck.svg.png" decoding="async" width="40" height="46" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/60px-Watchduck.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Watchduck.svg/80px-Watchduck.svg.png 2x" data-file-width="703" data-file-height="806"></a>
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<td><a href="//commons.wikimedia.org/wiki/User:Watchduck" title="User:Watchduck">Watchduck</a><br><span style="color: gray; font-size: 80%;">You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".</span>
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<span class="int-own-work" lang="en">Own work</span>