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An operating-system process's working set is the set of distinct memory pages used in a declared recent window. Here the window counts memory references, not clock time: let $r_i$ identify the memory page accessed by the $i$th memory reference, and let the positive integer $\Delta$ be the window length. Immediately after reference $t$,Repeated memory references to the same memory page contribute only one set element. The working set size is the cardinality $|W(t,\Delta )|$. This recent footprint estimates current demand; it cannot guarantee which memory pages future memory references will need. The resident set instead records which memory pages are actually present in computer memory.

For the 20-reference sequence in memory thrashing and $\Delta =4$, the successive working sets begin $\{1\}$, $\{1,2\}$, $\{1,2,3\}$ and $\{1,2,3,4\}$. At $t=5$, the window is references 2–5, namely $2,3,4,1$, so $W(5,4)=\{1,2,3,4\}$ again. Every later four-reference window contains all four memory pages. Under three-page frame LRU replacement, however, the resident set at $t=5$ is $\{1,3,4\}$: page 2 was evicted. Consequently the working set size is four even though only three memory pages are resident. Under four-page frame LRU replacement, the resident set contains the complete working set after reference 4.

Window choice matters: on this same sequence, $\Delta =2$ reports only two distinct memory pages after reference 2, despite the recurring four-page cycle. A short window forgets useful memory pages sooner; a long window retains evidence from older phases. Enlarging the window can only enlarge or preserve the working set at fixed $t$, because the shorter window is contained in the longer one. No single window is best for every workload.

For a concrete multiprogramming budget, suppose three operating-system processes each have a four-page working set, no shared page frames, and only eight page frames are available for their data. Their combined demand is 12 page frames; all three working sets cannot be resident together. Letting two operating-system processes run while the third waits makes their eight-page demand fit in this simplified model. Real budgets also reserve computer memory for the operating system and other uses; shared memory pages must not be counted repeatedly as different physical copies.

Peter J. Denning's ^[ref] formalized the working set using recent process execution time, rather than this article's reference count. Its central insight connected memory demand to process scheduling: deciding which operating-system processes run also decides which working sets need room. Temporal locality makes recent use informative, without making it a promise about the future. The beautiful distinction is between a large possible address space and the smaller footprint of the current activity.

Zero padding extends a finite sequence by zero entries to a specified length. For input lengths $r,s$, their full discrete convolution has length at most $r+s-1$. Padding both inputs to $N\ge r+s-1$ allows circular convolution to represent this full result without folded entries.

Let $(P,\le )$ be a nonempty partially ordered set. If every chain in a partial order $C\subseteq P$ has an upper bound in $P$, then $P$ has a maximal element. The hypothesis meansThe upper bound may depend on $C$ and need not belong to $C$. A least upper bound is not required. The conclusion asserts that some $m\in P$ satisfies $m\le x⟹x=m$; it does not assert a greatest element, which would satisfy $x\le m$ for every $x\in P$.^[ref]

Here a chain in a partial order may be the empty set. Every member of $P$ is an upper bound of the empty chain in a partial order, so including it in the hypothesis already forces $P$ to be nonempty. If “chain” means only nonempty chains in a partial order, the separate nonempty-$P$ assumption is essential: the empty poset otherwise satisfies the hypothesis vacuously and has no maximal element. These are two equivalent conventions for the theorem.

The strength of Zorn's lemma is that compatible partial constructions can have a completed, unextendible member without an explicit procedure for finding it. The next examples separate the upper bound calculation from the reason a maximal element has the desired property. All chains in a partial order must be covered, including uncountable ones, rather than just increasing sequences. The condition is sufficient, not necessary: a poset can have a maximal element and still have an unbounded chain in a partial order elsewhere.